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

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 emission 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 photons 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: � �

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 parallel,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 excitation 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

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 synchrotron 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 densitiesn 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 . � crit Collisions with protons and electrons can also be important for hyperfine excitation of H I; the rate coefficient for deexcitation by electrons is (Furlanetto & Furlanetto 2007)

9 0.5 3 1 k10(e−) 2.26 10 − (T/100 K) cm s− (1 < T < 500 K), (17.14) ≈ × ∼ ∼ a factor 10 larger thank (H); electrons will, therefore, be of minor importance ∼ 10 COLLISIONAL EXCITATION 195 in regions of fractional ionizationx e < 0.03, such as the CNM or WNM (see Figs. 16.1 and 16.2). ∼ Resonant scattering of Lyman-α photons can also change the populations of the 3 hyperfine levels (Wouthuysen 1952; Field 1958). LetP 01 be the probability per unit time of a transition from the hyperfine ground state to2p, followed by spon- taneous decay from2p to the hyperfine excited state, andP 10 the probability per time of a transition from the hyperfine excited state to2p, followed by decay to the hyperfine ground state. The Lymanα profile depends on the kinetic temperature of the hydrogen atoms that are emitting and scattering the Lymanα. If the hydrogen atoms have a Maxwellian velocity distribution, Field (1959) showed that 0.0682 K/T 2 P 3P e− H , whereT m σ /k, andσ is the one-dimensional 01 ≈ 10 H ≡ H V V velocity dispersion of the H atoms that are scattering the Lymanα photons. There- fore, this process acts essentially like a collisional process, except that the tem- peratureT H characterizing the Lymanα line profile includes a contribution from turbulent motions, in addition to microscopic thermal motions. Liszt (2001) estimates the Lymanα intensities expected in the warm neutral medium (WNM), and concludes that collisions and Lymanα together are not fast enough to thermalize the H I hyperfine transition. As a result, we should expectT spin < Tgas in the WNM.

17.4 Example: C II Fine Structure Excitation

The ground electronic state1s 22s22p 2P o of C II contains twofine-structure lev- 2 o 2 o els (see Fig. 17.3), P1/2 and P3/2. Will the populations of these two levels be 2 o thermalized in the ISM? Radiative decay of the P3/2 excitedfine-structure state produces a photon withλ = 158µm. At this wavelength, the continuum background in the interstellar medium hasn γ 1. In fact, from Figure 12.1, we 5 � estimaten 10− in the diffuse ISM. Hence, if we are considering regions that γ ≈ are optically thin in the 158µm line, we can neglect stimulated emission. 2 o 2 o The P3/2 fine-structure level can be excited by collisions of P1/2 with electrons, H, He, and (in a molecular cloud) H2. For electrons, the collision strength is (Keenan et al. 1986) Ω(2P o ,2 P o ) 2.1, (17.15) 1/2 3/2 ≈

3 The Wouthuysen-Field effect can be understood semiclassically. When a Lymanα photon is absorbed, the H atom enters a2p state with its electronic angular momentumL in some direction that is related to the direction of propagation and polarization of the absorbed photon, but is unrelated to the orientation of the nuclear or electron spins. During the 10 9 s lifetime of the 2P o or 2P o excited ∼ − 1/2 3/2 state, spin-orbit coupling will cause both the electron spinS and orbital angular momentumL to precess aroundL+S. When the Lymanα photon is emitted, the electron spin will be in a different direction, and thus its orientation relative to the nuclear spin will change some fraction of the time. Note that 1 while the spin-orbit coupling in H is weak, it is not zero: the0.366 cm − ﬁne-structure splitting between 2P o and 2P o corresponds to an electron precession frequency 8 10 10 Hz – the 1.6 ns 3/2 1/2 ∼ × ∼ lifetime of the excited state corresponds to 10 2 precession periods. ∼ 196 CHAPTER 17

Figure 17.3 Fine-structure levels of C+.

2 o + Figure 17.4 Excitation of the P3/2 excitedﬁne-structure level of C . The background 5 radiation is assumed to haven γ 10− at 158µm. ≈ so that

8 1/2 3 1 k (e−) 4.53 10 − T − cm s− , (17.16) 10 ≈ × 4 while for H atoms (Barinovs et al. 2005):

10 0.1281+0.0087 lnT 2 3 1 k (H) 7.58 10 − T cm s− . (17.17) 10 ≈ × 2 COLLISIONAL EXCITATION 197

Thus the critical densities are

3 0.1281 0.0087 lnT 2 3 n (H) 3.2 10 T − − cm− , (17.18) crit ≈ × 2

1/2 3 n (e−) 53T cm− . (17.19) crit ≈ 4 Therefore, we see that for both CNM and WNM conditions, the densities are well below critical, and the C IIﬁne-structure levels will be subthermally excited. It fol- 2 o lows that collisional excitations of C II P3/2 will usually be followed by radiative decays, removing energy from the gas. The [C II] 158µm transition is the principal cooling transition for the diffuse gas in star-forming galaxies. Plate 3c is an all-sky map of [C II] 158µm emission from the Galaxy, made by the Far InfraRed Ab- solute Spectrophotometer (FIRAS) on the COsmic Background Explorer (COBE) satellite (Fixsen et al. 1999). 2 o The preceding discussion neglects radiative excitation of C II P3/2, appropriate for clouds that are optically thin in the [C II] 158µm line. When the clouds become optically thick, the [C II] 158µm line intensity can increase to the point where self- 2 o absorption makes an important contribution to the excitation of C II P3/2. We will return to the question of excitation under such conditions in Chapter 19.

17.5� Three-Level Atom

If we consider the ground state and two excited states, we refer to this as a “three- level atom.” The equations for the evolution of the level populations are dn 2 =R n +R n (R +R )n , (17.20) dt 02 0 12 1 − 20 21 2 dn 1 =R n +R n (R +R )n , (17.21) dt 01 0 21 2 − 10 12 1

where the ratesR if are:

R10 =C 10 + (1 +n γ,10 )A10 , (17.22)

R20 =C 20 + (1 +n γ,20 )A20 , (17.23) R21 =C 21 + (1� +n γ,21 )A21 , � (17.24) E10/kT R01 = (g1/g0) C10e− +n γ,10 A10 , (17.25) � � E20/kT R02 = (g2/g0) C20e− +n γ,20 A20 , (17.26) � � E21/kT R12 = (g2/g1) C21e− + (g2/g1)nγ,21 A21 . (17.27)

The ratesC u� for collisional deexcitation are summed over all collision partnersc: � C n k (c), (17.28) u� ≡ c u� c 198 CHAPTER 17 and we assume each colliding species to have a thermal velocity distribution corresponding to temperatureT . For the three-level system, the solution is tractable:

n R R +R R +R R 1 = 01 20 01 21 21 02 , (17.29) n0 R10R20 +R 10R21 +R 12R20 n R R +R R +R R 2 = 02 10 02 12 12 01 . (17.30) n0 R10R20 +R 10R21 +R 12R20

For systems with more than three energy levels, the steady state level populations are usually found using numerical methods to solve the system of linear equations.

17.6� Example: Fine Structure Excitation of C I and O I

CI(1s22s22p2) and O I (1s22s22p4) are two important examples of atoms with triplet (S=1) ground states (see Fig. 17.5). Using collisional rate coefﬁcients from Appendix F, we can solve for excitation of these levels. Results forn 1/n0 for C I are shown in Fig. 17.6, andn 1/n0 for O I are shown in Fig. 17.7. For both 3 cases, we have assumed a fractional ionizationn e/nH = 10− characteristic of H I clouds or photodissociation regions.

Figure 17.5 Fine-structure levels of C I and O I.

17.7� Measurement of Density and Pressure Using C I

Theﬁne-structure excited states of C I, with energiesE 1/k = 23.6 K andE 2/k= 62.5 K, can be collisionally populated even at low temperatures, and the level populations can be measured using C I’s rich spectrum of ultraviolet absorption lines COLLISIONAL EXCITATION 199

3 Figure 17.6 Excitation of C I P1, source of 609.1µm emission.

3 Figure 17.7 Excitation of O I P1, source of 63.18µm emission.

(see Appendix E). Because the critical densityn H,crit (see Table 17.1) is higher than the densities in typical diffuse clouds, the population of the C Iﬁne structure levels can be used to constrain the density and temperature (Jenkins & Shaya 1979). A recent study by Jenkins & Tripp (2010) used high-quality spectra of UV absorption lines of C I on 89 sightlines to characterize the distribution of thermal pressures in diffuse clouds.

For a given gas composition (fractional ionization andH 2 fraction), temperature T , and density, the fractionsf N( 3P )/N(C I) andf N( 3P )/N(C I) of 1 ≡ 1 2 ≡ 2 200 CHAPTER 17

3 3 the C I that are in theﬁrst and second excited states P1 and P2 of the ground 3 electronic state P can be calculated theoretically (e.g., Fig. 17.6 showsf 1/f0). For a givenT , varying the thermal pressurep will generate a track in thef 1–f2 plane. Theoretical tracks forT = 30 K, 80 K, and 300 K are shown in Figure 2 3 17.8, with each track generated by varying the pressure from p/k = 10 cm− K 7 3 to 10 cm− K.

3 3 Figure 17.8f1 andf2 are the fractions ofCI that are in the P1 and P2 excitedﬁne structure states. Solid lines are theoretical tracks for three different temperatures (30 K, 2 3 7 3 80 K, 300 K), as the pressure is varied from p/k = 10 cm− K to 10 cm− K, with 3 numbers indicating the value of log10[p/k( cm− K)]. Data points are measurements for different velocity components on 89 sightlines. The area of each dot is proportional toN(C I). The white is the “center of mass” value(f1, f2) = (0.21,0.068). Taken × from Jenkins & Tripp (2010).

Observed values of(f , f ) are also plotted in Figure 17.8. Typicallyf 1 2 1 ≈ 0.20 of the C I is found to be in the 3P level, andf 0.07 is in the 3P level, 1 2 ≈ 2 although on some sightlinesf 2 can exceed 0.50. Note that the observed values of(f 1, f2) usually fall somewhat above and to the left of the theoretical tracks. Jenkins & Tripp (2010) interpret this tendency as resulting from superposition of two components: a dominant component with moderate pressurep plus a small COLLISIONAL EXCITATION 201 amount of high-pressure material.4 UV pumping can also populate the excited ﬁne-structure levels, and Jenkins & Tripp (2010) correct for this in their estimates forp.

Figure 17.9 Distribution of thermal pressures measured using C I absorption lines (see text). Taken from Jenkins & Tripp (2010).

The distribution of pressures found by Jenkins & Tripp (2010) is shown in Fig. 17.9. The observed distribution can be approximated by a log-normal distribution 3.575 3 3 with a peak at p/k 10 cm− K 3800 cm − K. ≈ ≈ In many cases it was possible to determine the gas temperature using theH 2 J=1 0 rotation temperatureT . The inset in Fig. 17.9 shows the distribution − rot ofp andT rot; there appears to be no correlation betweenT rot andp. For the 3 “typical” p/k 3800 cm − K, theH rotation temperatures range from 50 K to ≈ 2 ∼ 250 K. ∼ Jenkins & Tripp (2010) conclude that interstellar clouds routinely contain a small amount of gas that is at the same bulk velocity but at a pressure that is much higher than the average pressure in the cloud – this is the only way that they can explain the tendency of the data points in Fig. 17.8 to fall above and to the left of the theoretical tracks. This is a very surprising result, as there is no obvious explanation for

4 Jenkins & Tripp (2010) assume the high-pressure material to have(f 1, f2) = (0.38,0.50). 202 CHAPTER 17 why a small fraction of the cloud material should be overpressured without being at a signiﬁcantly different velocity. It may be conjectured that the overpressured regions are the result of highly localized intermittent heating, perhaps due to turbulent dissipation, but the situation remains unclear. It is at least conceivable that the problem could be with the collisional rate coefﬁcients – if, for example, the cur- rent theoretical rates have too small a value ofC 20/C10, then the true tracks at the low-pressure end of Fig. 17.8 would have a larger slope, perhaps passing through the cloud of points near (0.2,0.06) in Fig. 17.8, and removing the need to invoke an admixture of high pressure material.