–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 − pdρ−1, (1) where dq is the heat added to the system per unit mass, ρ−1 the specific volume (i.e., volume per unit mass), 1 p e = + e (2) γ − 1 ρ I the specific energy, γ the ratio of specific heats, and eI the specific 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 modified 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 = ρ nαIα, where α labels the ion species and Iα is the chemical/ionization energy per particle. For convenience, we define 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 µ be the mass per particle so that ρ = nµ, where n is the number density of particles, and p = nkT = ρkT/µ. Converting equation (1) to a rate equation and factoring out p/ρ = kT/µ, we find dq p 1 d p 1 de = ln + d ln + I , (3) dt ρ γ − 1 dt ρ ρ dt · µ ¶ µ ¶¸ kT d p de = ln + I , (4) µ(γ − 1) dt ργ dt µ ¶ where d/dt = ∂/∂t + v · 5 is the comoving, or Lagrangian, derivative. The quantity ρdq 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 ∝ ρ . Since heating processes are generally per particle, we define the heating rate per unit volume 1 as nHΓ. Cooling processes generally involve collisions between two particles, so they scale as the

1 The mass density ρ can also be written in terms of the number density of hydrogen nuclei as ρ = nHµH, where µH −24 is the mass per hydrogen nucleus. For a He abundance that is 10% that of hydrogen, µH = 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 ﬁxed volume, whereas n changes. Equivalently, µH is a constant in an ionizing gas, but µ changes. – 2 –

2 2 square of the density; we therefore write the cooling rate per unit volume as nHΛ. The total rate at which heat is added to the gas per unit volume is then dq p d p de ρ = n Γ − n2 Λ = ln + ρ I . (5) dt H H γ − 1 dt ργ dt µ ¶ 5 For γ = 3 and constant composition, this simpliﬁes to d 3 dn n kT − kT = n Γ − n2 Λ, (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

Γ = nHΛ (steady state). (7)

It is sometimes convenient to describe the evolution of the thermal state of the gas in terms of the specific 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 specific 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 (µ, eI const), equation (4) can be integrated to give the specific entropy of the gas, k T 1/(γ−1) s = ln + const. (8) µ " 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, nHΓ = 4πκJ:

- 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 Wolﬁre et al 1995).

2 Note that some people redeﬁne Γ and Λ 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 diﬀerent 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, determine 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 Γ v(`)2/2 v(`)3 ∼ = . (9) µH `/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 ﬁrst 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 significant heat source. In ionized components of the ISM, the field is “frozen” to the plasma and cannot directly heat it. However, on sufficiently small scales, flux freezing breaks down and the topology of the field 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 diffusion: The magnetic field 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 significant, as it can be in the weakly ionized gas in molecular clouds, then this process can lead to significant 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) coefficient is written as αA or α , indicating that recombinations to all states are effective. 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. The recombination coefficient in this case is written as αB or α(2). Note that if the energy of the recombining electron is less than the average, the temperature of the gas will actually increase as a result of this cooling. (Correspondingly, if a photoionization produces an electron with an energy lower than the average, it will lower the temperature of the gas.) - Recombination cooling on dust. Electrons can also recombine onto dust grains, emitting energy and cooling the gas.

Bremsstrahlung (free-free).

- Collisions between free electrons and ions produce bremsstrahlung, which is also called free-free emission in astrophysics. A signiﬁcant fraction of the radio and infrared emission from H II regions is free-free emission. – 5 –

5.2. Two-Level Atom

To determine the rate of collisional cooling, it is useful to introduce the simple model of a two- level atom. Real atoms and molecules are much more complex, but this simple model introduces some of the basic concepts of collisional cooling.

For clarity, we label the lower state by l and the upper state by u, so that the Einstein coeﬃcients connecting the two levels are Aul, Bul, and Blu. The rate at which electron collisions excite the atom to the upper level is neγlu (this treatment can be easily generalized to include atomic or molecular collisions). Electrons can also de-excite the atom at a rate neγul. In a steady state, the rate of excitation balances the rate of de-excitation,

nu(Aul + Buluν + neγul) = nl(Bluuν + neγlu). (10)

In the particular case of thermodynamic equilibrium, the radiative rates balance separately, as we showed in Lecture 2. It follows that the collisional rates also balance: ∗ γlu nu gu −Elu/kT = ∗ = e . (11) γul nl gl The fact that the rates of individual processes are in balance in thermodynamic equilibrium is referred to as the Principle of Detailed Balance. The de-excitation rate is often written in terms of the collision strength Ωlu, which is approximately independent of T :

−6 Ωlu 3 −1 guγ = 8.63 × 10 cm s . (12) ul T 1/2 Usually the collision strength is of order unity, even for forbidden transitions.

In the ISM, the radiation ﬁeld is generally very weak. The energy density is comparable to that of the microwave background at TCMB ' 2.7 K, but the average energy of a photon in the ISM is several eV. The energy density of optical and UV radiation in the ISM is thus many orders of magnitude less than that of a blackbody at a temperature comparable to that of the average photon. For Elu/k À 3 K, the steady-state population of a two-level atom in the typical interstellar radiation ﬁeld is governed by n n γ u = e lu , (13) nl Aul + neγul γ 1 = lu , (14) γul 1 + Aul/neγul g 1 = u e−Elu , (15) g 1 + n /n l µ cr e ¶ where the critical density is Aul ncr ≡ . (16) γul 2 For low densities (ne ¿ ncr), the density of the upper state varies as nenl ∝ n and correspondingly 2 the emission rate varies as nuAul ∝ n . On the other hand, for high densities (ne À ncr), the – 6 – population of the upper state approaches its LTE value relative to the lower state (see eq. 11); the density of the upper state therefore varies as nl ∝ n and the emission rate varies linearly with n. Correspondingly, the excitation temperature approaches the gas temperature, Tex → T .

Consider some numerical examples. For a collision strength Ωlu ∼ 1 and a temperature T ∼ 4 −7 3 −1 10 K, the collisional de-excitation rate is γul ∼ 10 cm s . Permitted optical and UV lines 8 −1 15 −3 have Aul ∼ 10 s , corresponding to ncr ∼ 10 cm , much greater than densities in the ISM. On the other hand, forbidden lines have much smaller transition probabilities, so that the critical −3 densities are in the range of interstellar densities. For example, ncr = 50 cm for [C II] λ 158 µm −3 and ncr = 3400 cm for [O II] λ 3729.

5.3. The Temperature of H II Regions

Recall that the temperature of a pure hydrogen H II region is T ' 1.5hψiT∗, where the average heating per photoionization is ψkT∗ and ψ = O(1). How is this result altered by the inclusion of cooling by collisional excitation of transitions in the heavier elements?

The heating rate per unit volume due to photoionization is (Lecture 4) 0 (2) + ζπn(H )hψikT∗ = α nen(H )hψikT∗, (17) since the rate of ionization is balanced by the rate of recombination to excited states. Because oxygen is the most abundant element after helium, it dominates the cooling. Figure 3.2 in Osterbrock and Ferland shows that about half the cooling is due to the forbidden lines of O II at 3726 A˚ and −3 3729 A.˚ Provided that ne ¿ ncr = 3400 cm , balancing heating and cooling gives (2) + + α nen(H )hψikT∗ ' 2nen(O )γluElu. (18) + Since the hydrogen in an H II region is highly ionized, we have n(H ) ' nH so that n(O+) g γ α(2)hψikT ' 2 u ul e−Elu/kT E , (19) ∗ n g lu · H ¸ l n(O+) 8.63 × 10−6Ω ' 2 lu e−Elu/kT E , (20) n g T 1/2 lu · H ¸ l where we used equation (11) to obtain the ﬁrst equation and equation (12) to obtain the second. 4 The ground state is a S state (see below) so that gl = 4. Osterbrock and Ferland give Ωlu = 1.34 for this transition. Outside the exponential, we approximate the temperature as T ∼ 104 K, whereas inside the exponential we have Elu/kT = 3.32 eV/kT = 38, 500 K/T . Recall that the abundance of oxygen is 4.6 × 10−4 (Lecture 1). Solving for T , we ﬁnd 38, 500 K T ' . (21) ln(390/hψiT∗4)

For example, a B0.5 star has T∗ ' 32, 000 K and hψi = 1.38, giving an H II region temperature T ' 8600 K. Since this depends only logarithmically on the stellar properties, we conclude that cooling by heavy elements regulates the temperature of H II regions to be about 104 K. – 7 –

5.4. Some Elements of Atomic Spectroscopy

Atomic spectroscopy is a rich and complex subject. Here we give a very abbreviated discussion of some of the terminology relevant for discussing emission lines in the interstellar medium; most of this is taken from Osterbrock and Ferland. Recall that each electron in an atom is characterized by its radial quantum number, n, which labels which shell it is in, and its angular momentum, l. Each shell can hold at most 2n2 electrons due to the Pauli exclusion principle. Thus, the K shell (n = 1), can hold at most 2 electrons; the L shell (n = 2) can hold at most 8; the M (n = 3) shell 18, etc. The electron configuration of an ion describes the shell and angular momentum of each electron. For example, the ground state of the 6-electron ion O++ is 1s2 2s2 2p2, indicating 2 electrons in the n = 1 shell with zero angular momentum, 2 electrons in the n = 2 shell with zero angular momentum, and 2 electrons in that shell with l = 1. The electronic configuration of the outermost electrons in the ground state of some of the ions of greatest astrophysical interest are: Table 5.1 Ground configurations

C N O Ne 2p C II N III 2p2 C I N II O III 2p3 N I O II 2p4 O I 2p5 Ne II

The abundant elements generally have low nuclear charge Z, and for these elements the orbital angular momenta of the individual electrons add to a total orbital angular momentum with quantum number L, and the spin angular momenta add to a total spin angular momentum with quantum number S (LS coupling). For high-Z elements, spin-orbit coupling dominates, so that the individual orbital and spin angular momenta add (ji = li + si), which in turn add to the total angular momentum J (jj coupling). Iron has an intermediate value of Z; its spectrum is better described by LS coupling than jj coupling. With LS coupling, an energy level is denoted by

2S+1 LJ , (22) J = L + S is the total angular momentum quantum number; here the “+” denotes the quantum mechanical addition of angular momenta. In spectroscopic notation, L = 0, 1, 2, 3 are denoted by the letters S, P, D, F, etc. The parity of the level is determined by the algebraic sum of the l-values of the electrons; if it is odd, a superscript “o” is added to the designation. For S ≤ L, there are 2S + 1 diﬀerent levels characterized by L and S; this set of levels is called a term, denoted 2S+1 ++ 3 L. For example, the ground state of the O ion (whose spectrum is denoted by O III) is P0, indicating that there are three levels in the ground term and that the level of lowest energy has J = 0. In a magnetic ﬁeld, levels break up into 2J + 1 magnetic sublevels. The statistical weight of an energy level is therefore gJ = 2J + 1; the statistical weight of a term is the sum of the gJ ’s, which can be shown to be gSL = (2S + 1)(2L + 1). – 8 –

Transitions between levels obey selection rules. Permitted (also called allowed) transitions are electric dipole transitions that obey the following rules:

- ∆J = 0, §1, except 0 → 0 is not allowed

- Parity changes

- One electron jumping with ∆l = §1 and ∆n arbitrary

- ∆S = 0 in LS coupling

- ∆L = 0, §1 in LS coupling

Transitions that do not obey these selection rules are called “forbidden.” (Transitions with ∆S 6= 0 are called “intercombination” or “semi-forbidden” since they are electric dipole transitions that violate LS coupling; they have oscillator strengths that are intermediate between allowed and forbidden transitions.) Table 5.2. Fine Structure and Optical Forbidden Lines Conﬁg. Term Ion 2p, 2p5 C II N III Ne II 2P o 158 µm 57 µm 12.8µm

2p2, 2p4 C I O I N II O III 1S – – 5755 4363 1D 9823, 9850 6300, 6364 6548, 6583 4959, 5007 3P 369µm 146µm 122µm 52µm 609µm 63µm 205µm 88µm

2p3 N I O II S II 2P o 7319-7331 2Do 5198, 5201 3726, 3729 6716, 6731 4So No fine str. Since transitions within an electronic configuration involve no change in the parity, they are forbidden; equivalently, permitted transitions require ∆l = §1, whereas ∆l = 0 for transitions within a given configuration. Transitions within a term are called fine structure lines; in such transitions, J changes but L and S remain fixed; the most important fine structure line in the ISM is [C II] λ158 µm, the dominant coolant in the cold neutral medium. Interaction with the magnetic moment of the nucleus leads to hyperfine structure; the most important astrophysical example of a hyperfine line is the 21 cm line of H0. Table 5.2 gives the wavelengths of some of the important optical forbidden lines and infrared fine structure lines in the ISM. For each configuration, the terms are listed in order of decreasing energy; the lowest term is the ground term. The full description – 9 – of a transition is given by the the (abbreviated) configuration and level of the initial and final 2 1 2 1 state. For example, the 5755 A˚ line of N II is 2p D2 − 2p S0 λ5755. Transitions between the uppermost term and the ground state are not included. Note the equivalence of the 2p and 2p5 configurations (one electron and one hole, respectively) and of the 2p2 and 2p4 configurations.

5.5. Inferring the Density in an H II Region

H II regions are very inhomogeneous, as the famous picture of the Eagle Nebula shows. Fur- thermore, as we shall see in a later lecture, the inner regions of H II regions are believed to be ﬁlled with hot, shocked stellar wind gas at very low density. Nonetheless, it is possible to infer the density of the photoionized gas in an H II region from observations of forbidden lines.

O II is a good density diagnostic for two reasons: First, it is possible to estimate the density from observation of O II lines alone, so that no corrections need to be made for abundances. Second, 3 4 o 3 2 o 3 4 o 3 2 o the two lines, the 2p S3/2 − 2p D3/2 λ3726 and 2p S3/2 − 2p D5/2 λ3729 lines are very close in wavelength, so that no correction for diﬀerential reddening is needed. Label the energy levels 1, 2, 3 in order of increasing energy, so that λ12 = 3729 and λ13 = 3726. At low densities −3 (ne ¿ ncr[O IIλ3729]=3400 cm ), we have j(λ3729) γ g γ /g Ω 3/5 3 = 12 = 2 12 1 = 12 = = , (23) j(λ3726) γ13 g3γ13/g1 Ω13 2/5 2 where we used Osterbrock and Ferland equation (3.21) to evaluate the ratio of the collision 4 −3 strengths. On the other hand, at high densities (ne À ncr[O IIλ3726] = 1.5 × 10 cm ), we have −5 j(λ3729) n2A21 g2A21 6 × 3.6 × 10 = = = −4 = 0.34. (24) j(λ3726) n3A31 g3A31 4 × 1.6 × 10 2 −3 4 −3 Since the ratio of the line intensities changes by a factor of about 5 for 10 cm . ne . 10 cm , it is possible to determine the density relatively accurately in this range.

5.6. Determination of the Temperature in an H II Region

To infer the temperature of an H II region, it is best to use an ion with three terms in the ground conﬁguration, such as O++. Generally, one must infer the electron density together with the temperature, but the critical density for the [O III] lines is about 105 cm−3. Visible H II regions usually have lower densities than this, enabling a pure temperature determination.

Recall that [O III] has a 3P term (the ground state), a 1D term and a 1S term. Label these as 1, 2, 3. In the low-density limit we have

1 A j = hν n n(O++) γ + γ 32 , (25) 21 4π 21 e 12 13 A + A µ 32 31 ¶ – 10 –

1 ++ A32 j32 = hν32nen(O )γ13 , (26) 4π A32 + A31 where the 2 → 1 transition is λλ4959, 5007 and the 3 → 2 transition is λ4363. The ratio A32/(A32 + 1 1 A31) = 0.89 is the branching ratio from S to D. It follows that

j /ν γ 32 32 = 0.89 13 , (27) j21/ν21 − j32/ν32 γ12 Ω = 0.89 13 e−(E13−E12)/kT . (28) Ω12 Since the line intensities are proportional to the emissivities, we ﬁnd

4363I(λ4363) ∝ e−(E13−E12)/kT = e−33000 K/T . (29) 4983I(λ4959 + λ5007) − 4363I(λ4363)

The exponential sensitivity makes the [O III] lines a good thermometer.