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Collisional Excitations of C II P3/2 Will Usually Be Followed by Radiative Decays, Removing Energy from the Gas

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Collisional Excitations of C II P3/2 Will Usually Be Followed by Radiative Decays, Removing Energy from the Gas Chapter Seventeen Collisional Excitation Collisional excitation is important in the ISM for two reasons: 1. It puts ions, atoms, and molecules into excited states from which they may decay radiatively; these radiative losses result in cooling of the gas. 2. It puts species into excited states that can serve as diagnostics of the physical conditions in the gas. If the level populations can be determined observation- ally, from either emission lines or absorption lines, we may be able to infer the density, temperature, or radiationfield in the region where the diagnostic species is located. Throughout this chapter and the rest of the book, we will be making use of rate coefficients and transition rates. We will use the notationk if σv i f ≡� � → for collisional rate coefficients, andA if A i f to denote radiative transition ≡ → probabilities, where thefirst subscript ink if orA if denotes the initial state, and the second thefinal state. For energy-level differences, we will setE E E . u� ≡ u − � 17.1 Two-Level Atom In some cases, it is sufficient to consider only the ground state and thefirst excited state – when attention is limited to these two states, we speak of the “two-level atom.” Considerfirst the case where there is no background radiation present, and the only processes acting are collisional excitation, collisional deexcitation, and radiative decay. Let the ground state be level 0, and the excited state be level 1. Letn j be the number density of the species in levelj. For collisional excitation by some species (e.g., electrons) with densityn c, the population of the excited state must satisfy dn 1 =n n k n n k n A . (17.1) dt c 0 01 − c 1 10 − 1 10 The steady state solution (dn1/dt=0) is simply n n k 1 = c 01 . (17.2) n0 nck10 +A 10 The upward rate coefficientk 01 is given in terms of the downward rate coefficient COLLISIONAL EXCITATION 191 by g1 E10/kTgas k01 = k10 e− , (17.3) g0 whereg 0,g 1 are the level degeneracies, andT gas is the kinetic temperature of the gas. In the limitn , it is easy to see thatn /n (g /g ) exp( E /kT ). c → ∞ 1 0 → 1 0 − 10 gas Now suppose that radiation is present. Letu ν be the specific energy density at frequencies nearν=E 10/h. It is convenient to use instead the dimensionless (angle- and polarization-averaged) photon occupation number: c3 ¯n u . (17.4) γ ≡ 8πhν 3 ν Then, � � dn1 g1 =n 0 nck01 + ¯nγ A10 n 1 [nck10 + (1 + ¯nγ )A 10]. (17.5) dt g0 − The rate of photoabsorption is¯nγ (g1/g0)A10n0, and the rate of stimulated emis- sion is¯nγ A10n1 (see Chapter 6). The steady-state solution with radiation present is n n k + ¯n(g /g )A 1 = c 01 γ 1 0 10 . (17.6) n0 nck10 + (1 + ¯nγ )A10 This is the fully general result for a two-level system. It is instructive to examine Eq. (17.6) in various limits: If¯n 0, then we recover our previous result (17.2). • γ → Ifn c 0, thenn 1/n0 (g 1/g0)¯nγ /(1 + ¯nγ ). If we have a blackbody • → → E10/kTrad radiationfield with temperatureT rad [i.e.,n γ = 1/(e 1)], then E10/kTrad − n1/n0 = (g1/g0)e− . If we have a blackbody radiationfield with temperatureT rad =T gas, then • E10/kTrad n1/n0 = (g1/g0)e− independent of the gas densityn c! The pho- tons alone are sufficient to bring the two level system into LTE, and additional (thermal) collisions have no further effect on the level populations. 17.2 Critical Densityn crit,u For a collision partnerc, we define the critical densityn crit,u(c) for an excited state u to be the density for which collisional deexcitation equals radiative deexcitation, 192 CHAPTER 17 Table 17.1 Critical Densities for Fine-Structure Excitation in H I Regions nH,crit(u) E�/k Eu/kλ u� T = 100 KT = 5000 K 3 3 Ion�u (K) (K)(µm) ( cm − ) ( cm− ) 2 o 2 o 3 3 C II P1/2 P3/2 0 91.21 157.742.0 10 1.5 10 3 3 × × CI P0 P1 0 23.60 609.7 620 160 3 3 P1 P2 23.60 62.44 370.37 720 150 3 3 5 4 OI P2 P1 0 227.71 63.1852.5 10 4.9 10 3 3 × 4 × 3 P1 P0 227.71 326.57 145.532.3 10 8.4 10 2 o 2 o × 5 × 4 Si II P1/2 P3/2 0 413.28 34.8141.0 10 1.1 10 3 3 × 4 × 4 Si I P0 P1 0 110.95 129.684.8 10 2.7 10 3 3 × 4 × 4 P1 P2 110.95 321.07 68.4739.9 10 3.5 10 × × including stimulated emission1: � �<u�[1 + (nγ )u�]A u� ncrit,u(c) . (17.7) ≡ �<u ku�(c) Note that the definition (17.7) applies to multilevel systems, but each excited level u may have a different critical density. The definition (17.7) is appropriate when the gas is optically thin, so that the radiated photons can escape. When the emitting gas is itself optically thick at the emission frequency, we have “radiative-trapping,” and the criterion for the critical density must be modified (see Chapter 19). Note that this definition ofn crit,u depends on the intensity of ambient radiation at frequencies where levelu can radiate. For many transitions of interest, we have (¯n) 1, and this correction is unimportant, but for radio frequency transitions γ u� � – e.g., the 21-cm line of atomic hydrogen – it is important to include this correction for stimulated emission. Critical densitiesn crit for thefine structure levels of C I, C II, O I, Si I, and Si II are given in Table 17.1. 17.3 Example: H I Spin Temperature Consider the ground state of the hydrogen atom (electron in the1s orbital, electron spin antiparallel to nuclear spin,g 0 = 1), and the hyperfine excited state (1s orbital, electron spin and nuclear spin parallel,g 1 = 3). The energy level structure is illustrated in Fig. 17.1. The energy difference between the excited state (nuclear and electron spins par- allel,g 1 = 3) and the ground state (nuclear and electron spins antiparallel) is only E10 = 5.87µeV, corresponding to a photon wavelengthλ = 21.11 cm. The spon- 15 1 taneous decay rate isA = 2.884 10 − s− , corresponding to a lifetime of 10 × 107 yr. ∼ 1The definition of critical density is not completely standard. Some authors include collisional exci- tation channels in the denominator of Eq. (17.7). COLLISIONAL EXCITATION 193 Figure 17.1 Hyperfine splitting of the1s level of H. The rate coefficient for collisional dexcitation of the hyperfine excited state due to collisions with other H atoms (Allison & Dalgarno 1969; Zygelman 2005) can be approximated by � 10 0.74 0.20 lnT 2 3 1 1.19 10 − T2 − cm s− (20 K<T< 300 K) k10 × 10 0.207 0.876/T 3 1 3 . (17.8) ≈ 2.24 10 − T e− 2 cm s− (300 K<T< 10 K) × 2 We obtaink 01 from the principle of detailed balance (3.21): 0.0682 K/T k01 = 3k10 e− . (17.9) What is the value of the photon occupation number¯nγ in the diffuse ISM? The radiationfield near 21 cm is dominated by the cosmic microwave background (CMB) plus Galactic synchrotron emission. Including the contribution2 from syn- chrotron radiation, the angle-averaged background near 21 cm corresponds to an antenna temperatureT T + 1.04 K (see Chapter 12), whereT = A ≈ CMB CMB 2.73 K. Thus 1 ¯n (17.10) γ ≡ exp(hν/kT ) 1 B − kT 3.77 K A 55. (17.11) ≡ hν ≈ 0.0682 K ≈ For the present case of a two-level system, (1 + ¯nγ )A10 ncrit(H) (17.12) ≡ k10 3 0.66 3 = 1.7 10 − (T/100 K)− cm− (50 K < T < 200 K). (17.13) × ∼ ∼ 2 Near 21 cm, even the CMB is in the Rayleigh-Jeans limit, so the antenna temperatureT A and brightness temperatureT B are approximately equal. 194 CHAPTER 17 where we have taken¯nγ = 55, appropriate for H I in the diffuse interstellar medium. In the absence of collisions, the CMB plus Galactic synchrotron radiation would give optically thin H I an excitation temperatureT 3.77 K. If we consider exc ≈ only collisions with atomic H, we can solve Eq. (17.6) for various gas densi- tiesn H and temperaturesT . Figure 17.2 shows the resulting “spin temperature” T 0.0682 K/ ln(n g /n g ) as a function of densityn . spin ≡ 0 1 1 0 H Figure 17.2 H I spin temperature as a function of densityn H, including only 21 cm continuum radiation (with brightness temperatureT B = 3.77 K, i.e.,n γ = 55) and collisions with H atoms. Lymanα scattering is not included. Filled circles shown crit for each curve. For densitiesn n , we expectT T , while for densitiesn n , � crit spin ≈ gas � crit we expectT T (ν), or 3.77K for the radiationfield assumed here. The spin ≈ B results in Fig. 17.2 are consistent with the expected asymptotic behavior, but it is important to note that one requiresn n in order to haveT within, say, � crit spin 10% ofT gas, particularly at high temperatures. The points in Figure 17.2 show ncrit(Tgas) for eachT gas; it is apparent that high values ofT spin are achieved only forn n .
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