Thermal Equilibrium in Nebulae 1 For an ionized nebula under steady conditions, heating and cooling processes that in isolation would change the thermal energy content of the gas are in balance, such that the gas is in thermal equilibrium, with a stable temperature. I. Heating. When a photon ionizes an atom, part of the photon s original energy hν is consumed in overcoming the atom s ionization potential χ i and the remainder is converted into kinetic energy of the liberated electron (what is referred to as a photoelectron ). This kinetic energy represents an addition to the thermal energy of the gas and the photoionization process itself thus acts as a heating process. The amount of photoelectric heating per unit volume will depend on the number density of atoms that are being ionized, the rate at which ionizing photons encounter those atoms (which will depend on the radiation intensity), the ionizing photon energies, and the crosssection for photoionization. For a pure hydrogen nebula, we can write the rate of heating, or energy Gained per unit volume per unit time, as Γ(H) = n(h 0 ) 4π I ν ν 0 hν h(ν ν 0 ) σ ν (H 0 ) dν, (1) where n(h 0 ) is the number density of neutral hydrogen, I ν is the specific intensity averaged over all directions, ν 0 is the photon frequency corresponding to the threshold for ionization (i.e., hν 0 = χ i = 13.6 ev), and σ ν (H 0 ) is the cross-section for hydrogen ionization as a function of frequency. Under steady conditions, the hydrogen at each point in the nebula will be in ionization equilibrium, which means that we can equate the photoionization and recombination rates: n(h 0 ) 4π I ν ν 0 hν σ ν (H 0 ) dν = n p n e α B (T ), (2) 1 Based on Osterbrock & Ferland (2006), Astrophysics of Gaseous Nebulae and Active Galactic Nuclei, 2nd Ed., Chapter 4. 1
so that n(h 0 )= and substitution into eq. (1) yields n p n e α B (T ) 4π I ν ν 0 hν σ ν (H 0 ) dν (3) Γ(H) = n p n e α B (T ) 4π I ν ν 0 hν h(ν ν 0 ) σ ν (H 0 ) dν 4π I ν ν 0 σ hν ν (H 0 ) dν. (4) Note the following: The ratio of integrals in eq. (4) is just the average kinetic energy per photoelectron; since photoionization is balanced by recombination, the rate at which photoelectrons are liberated per unit volume is given by n p n e α B, and the combined product in eq. (4) thus gives the heating rate per unit volume. Hotter stars produce higher energy photons that will lead to an increase in average kinetic energy per photoelectron, and hence a larger Γ(H), all other things being equal. When comparing the ionizing spectrum from two different stars, a common description is that the hotter star produces a harder spectrum, i.e. the average photon energy is larger, with correspondingly greater photoelectric heating. The full heating rate Γ will include the contributions of photoelectric heating from ionization of helium, Γ(He), and heavier elements, in addition to hydrogen. Physically these contributions will also be described with expressions like eq. (4), modified to include the appropriate bookkeeping for ionization. II. Cooling. Cooling occurs when thermal energy in the gas particles is converted to radiation which escapes from the nebula. The net result is that thermal energy is extracted from the gas. Several processes are important for cooling. A. Energy Loss by Recombination. When a free electron is captured to a bound state in an atom, a photon is emitted that carries away the free electron s kinetic energy plus 2
the binding energy corresponding to the state it is captured to. The rate at which energy is Lost from the plasma in this process per unit volume is thus Λ R (H) n p n e α B (H 0,T) kt. (5) Note the following: Equation (5) is an approximation. The average kinetic energy per particle in the gas is 3 2kT, but the slower particles are preferentially captured, so that the kinetic energy converted to radiation is somewhat less than the average value; here we represent it by kt. In this treatment of cooling as well as heating (eqs. 2 4), we assume Case B applies and neglect recombinations to the ground state. Equation (5) can be modified to estimate the contributions to recombination cooling by helium, Λ R (He), and by heavier elements. Since the photoionization heating and recombination cooling processes scale with the density of the ion, for cosmic abundances elements heavier than helium can generally be neglected. B. Energy Loss by Free-Free Radiation. In an ionized gas, an electron will radiate when it undergoes acceleration through Coulomb interaction with an ion. The emergent radiation is known as bremsstrahlung or free-free emission (reflecting the fact that the radiating particle is in an unbound state both before and after the emission occurs). The resulting rate of cooling by this process by ions of charge Z, integrated over all frequencies, is given by Λ FF (Z) =ε FF = 25 πe 6 Z 2 3 3/2 hm e c 3 ( ) 1/2 2πkT g ff n + n e, (6) m e where n + is the ion density and g ff is the mean Gaunt factor for free-free emission; the Gaunt factor is a dimensionless quantity that parameterizes quantum mechanical effects, 3
and has a value of order unity. An average value appropriate for nebular conditions is g ff 1.3. Plugging in values for the constants in eq. (6) yields Λ FF =(1.42 10 27 erg s 1 cm 3 ) Z 2 ( T K) 1/2 g ff ( n+ cm 3 )( ne ) cm 3. (7) Note the following: For a typical case of ionized hydrogen and singly ionized helium, both have Z =1so that n + = n p + n He II. Heavier elements can again be generally neglected due to their small abundance. Mass of the radiating particle appears in the denominator of eq. (6), which explains why electrons dominate free-free emission rather than protons or other more massive particles. C. Energy Loss by Collisionally Excited Line Radiation. Cooling in H ii regions is often dominated by spontaneous line emission from heavy element ions that have been collisionally excited. In this case thermal energy of the colliding particle, usually an electron, is transferred to the internal energy of the excited ion, and subsequently removed from the nebula by the emitted photon. For an atom with two energy levels, the cross-section for collisional excitation by an electron moving with velocity v is σ 12 (v) = π h2 m 2 e v2 Ω(1, 2) g 1 for 1 2 m e v 2 >χ 12 ; (8) here Ω(1, 2), known as the collision strength, is a dimensionless quantity that parameterizes the quantum mechanical effects and is a function of v; g 1 is the statistical weight of the lower level; and χ 12 is the excitation potential, i.e. the energy required for excitation from the lower level to the upper level. From LTE considerations, it is possible to derive a very similar expression for collisional deexcitation, in which case a transition occurs from level 4
2 to level 1, and the excitation energy χ 12 is added to the kinetic energy of the colliding electron: σ 21 (v) = π h2 m 2 e v2 Ω(1, 2) g 2. (9) Note that for a given pair of levels the same collision strength applies for both the excitation and deexcitation processes. The total collisional deexcitation rate per unit volume will depend on the density of target particles (ions in level 2), the density of colliding particles (electrons), the speed at which the colliding particles are moving around, and the cross-section for deexcitation, with an integration over velocities since not all particles are moving at the same speed. The total rate of collisional deexcitations per unit volume per unit time is thus n 2 n e q 21 = n 2 n e vσ 21 f(v) dv, (10) where q 21 is the deexcitation coefficient and f(v) is the distribution function for particle speeds (i.e, the Maxwellian distribution). Carrying out the integration in eq. (10) yields ( ) 1/2 2π h 2 n 2 n e q 21 = n 2 n e kt me 3/2 where Υ(1, 2) is the velocity-averaged collision strength, so that 0 Υ(1, 2) g 2, (11) q 21 =(8.629 10 6 cm 3 s 1 ) ( ) 0.5 T Υ(1, 2). (12) K g 2 The Boltzmann equation enables us to relate the coefficients for excitation q 12 and deexcitation q 21, with the result that q 12 = g 2 g 1 q 21 exp( χ 12 /kt ) (13) and q 12 =(8.629 10 6 cm 3 s 1 ) ( ) 0.5 T Υ(1, 2) exp( χ 12 /kt ). (14) K g 1 5
Some Υ values for transitions of interest are tabulated by Osterbrock & Ferland in their Tables 3.6 and 3.7. Note that these tables use a compressed representation for instances where no j quantum number is indicated. These cases correspond to transitions between a term consisting of a single level (i.e. single j quantum number) and a term with several levels distinguished by different j quantum numbers. In this case, the tabulated values represent the sum of collision strengths for the different possible transitions between the two terms. The Υ value for a single transition can be obtained from the relation Υ(SLJ,S L J )= (2J +1) (2S + 1)(2L +1) Υ(SL,S L ), (15) where the primed quantum numbers correspond to the level of interest in the multi-level term and the unprimed quantum numbers correspond to the level in the single-level term. The summed collision strength, Υ(SL,S L ), corresponds to the value listed in the tables. Note the following: The energy level structure for an atomic system is a function of the ionization state, and not simply the atomic number. Ions that are isoelectronic (such as N ii, O iii, Ne v) have similar level structures shifted in energy by the differing nuclear charge. Ions with the same number of outer shell electrons (such as O ii and S ii, which fall in the same column of the periodic table) will also have similar level structures. For common ions in nebulae, transitions between the low-lying excited states and the ground state violate selection rules for electric dipole transitions, but radiative deexcitations can still occur via magnetic dipole or electric quadrupole transitions. The quantum mechanical probabilities for such forbidden transitions are small, with the result that the Einstein A values are small and the deexcitation process tends to be slow; Einstein A values for transitions of interest are tabulated in Osterbrock & Ferland Tables 3.12 and 3.13. 6
Forbidden transitions are indicated in notation by including brackets around the ion; for example [O iii] λ5007, where 5007 indicates the wavelength in Å. Transitions that violate only the S selection rule are referred to as intercombination, orsemiforbidden transitions, indicated in notation by a single bracket on the ion: e.g. C iii] λ1909. Under steady conditions the number of ions in a given energy state will obey an excitation equilibrium, such that the rate of transitions into a given state will equal the rate of transitions out of it. For our two-level atom under nebular conditions, the upper state is populated by collisions and depopulated by collisions as well as spontaneous radiative transitions. In mathematical terms, this means that n 1 n e q 12 = n 2 n e q 21 + n 2 A 21, (16) from which it follows that the density of ions in the upper state is [ ] n e q 12 1 n 2 = n 1 A 21 1+ n e q 21. (17) A 21 The emitted photons remove energy from the nebula, and the cooling rate is thus Λ C = n 2 A 21 hν 21 (18) so that [ ] 1 Λ C = n 1 n e q 12 hν 21 1+ n e q 21. (19) A 21 Under interstellar conditions, particle densities are often very low. In the limit as n e 0, eq. (19) reduces to Λ C = n 1 n e q 12 hν 21. (20) In this low density limit, every collisional excitation is followed by emission of a photon. In the limit of high density, with n e, eq. (19) becomes Λ C n 1 n e q 12 hν 21 n e q 21 /A 21 = n 1 g 2 g 1 exp( χ 12 /kt ) A 21 hν 21. (21) 7
Comparison of eq. (21) with eq. (18) implies that n 2 = n 1 g 2 g 1 exp( χ 12 /kt ), (21) which is just the Boltzmann equation; in the high density limit a collisional equilibrium applies, with collisions dominating both excitations and deexcitations. In a multi-level atom, we can calculate the total line cooling by summing expressions like eq. (18) for each transition of interest, after first solving for the number densities of the excited states. The level populations will in general be set by an excitation equilibrium involving both radiative and collisional transitions, which for level i implies n j n e q ji + n j A ji = n i n e q ij + n i A ij. (22) j i j>i j i j<i The excited state densities sum to the total density of the ion: n j = n. (23) j Combining the equations (22) and (23) allows solution of the level populations; in practice only a limited number of states (often just the bottom 5) are considered in carrying out the solution. The collisionally excited radiative cooling rate for a given ion is then given by Λ C = i n i A ij hν ij. (24) j<i In the low-density limit, n e 0, this becomes a sum of terms like eq. (20). However, if the density is increased, collisions become important and a more complete solution is required. It is customary to define the critical density for a given level as the density where collisional removal from the state becomes equal to the radiative de-excitation rate (which may involve more than one transition): n crit (i) = j<i A ij j i q ij. (25) 8
See Table 3.15 from Osterbrock & Ferland for n crit values for some levels of interest. Thermal Equilibrium In a stable nebula, heating matches cooling, which means that Γ=Λ=Λ R +Λ FF +Λ C. (26) Here Λ C represents the sum of collisionally excited line emission from the different ions present in the nebula. It is worth noting how each of the terms in eq. (26) depends on the gas temperature. Γ(H), given by eq. (4), depends on T through the recombination coefficient α B (T ); recall that (approximately) α B T 0.7, which means that Γ is a decreasing function of temperature. From eq. (5), Λ R α B kt T +0.3,anincreasing function of temperature. From eq. (6), Λ FF T 1/2,anincreasing function of temperature. Since collisional excitations increase with increasing temperature, Λ C is an increasing function of temperature. (See eqs. 20 and 21 for behavior in the low and high density limits, with q 12 given by eq. 14.) Since each of the relevant cooling processes is an increasing function of T,theirsumΛ is likewise an increasing function of temperature. Since Γ(T ) is a monotonically decreasing function and Λ(T ) is a monotonically increasing function, eq. (26) is satisfied at a single value of T, which will be the equilibrium temperature for the nebula. See Figures 27.1 27.3 from Draine for illustrative cases (in his notation Γ pe = photoelectric heating, Λ rr = recombination cooling, Λ ce = cooling by collisionally excited emission). Note the following: 9
Given the important and often dominant role of collisionally excited cooling by ions of heavy elements, it is not surprising that the equilibrium temperature will be sensitive to the nebular abundances. As the abundance of heavy elements, or metallicity, increases, the larger density of coolants leads to an increase of Λ(T ) and a decrease in equilbrium temperature. A counter-intuitive result is that increasing the nebular abundance can result in a decrease in the strength of optical line emission from the nebula, as the cooling becomes increasingly dominated by infrared fine structure lines that have a weak sensitivity to temperature. Fine structure lines result from transitions within the ground-state term of an ion that differ in J quantum number. Increasing the nebular density will decrease the efficiency of radiative deexcitations relative to collisional deexcitation, with the result that Λ(T ) gets smaller and a hotter equilibrium T will result. The emissivity of a specific line is impacted when the nebular density n exceeds the transition s critical density n crit (see eq. 25). 10