University of Heidelberg, Center for Astronomy Dimitrios A. Gouliermis & Ralf S. Klessen Lecture #7 Physical Processes in Ionized Hydrogen Regions Part II (tentative) Schedule of the Course Lect. 1 Lect. 2 Lect. 3 Lect. 4 Lect. 5 Lect. 6 Lect. 7 Lect. 8 Lect. 9 Lect. 10 Lect. 11 Lect. 12 Lect. 13 19-Oct-2012 Course Overview Motivation for the Course/Schedule; Overview of Physical Processes in HII Regions; Classification of HII regions 26-Oct-2012 Introduction to the Physics of the ISM I Phases of the ISM; Transitions; Introduction to cooling mechanisms 2-Nov-2012 Introduction to the Physics of the ISM II Atomic Transitions; Gas Cooling; Collisional Excitation 9-Nov-2012 Introduction to the Physics of the ISM III Gas Heating; Photo-ionization; Photo-electric heating; PAHs 16-Nov-2012 Interstellar Dust Composition, Spectral Features, Grain Size Distributions, Extinction 23-Nov-2012 Physical Processes in HII Regions I Radiative Processes; Photo-ionization & Recombination of hydrogen; Photoionization Equilibrium 30-Nov-2012 Physical Processes in HII Regions II Heating and Cooling of HII Regions; Strömgren Theory; Forbidden lines and Line Diagnostics 7-Dec-2012 Photodissociation regions (PDR) Ionization & Energy Balance; Dissociation of Molecular Hydrogen; Structure; Observations 14-Dec-2012 Stellar Feedback Processes Dynamics of the ISM; Ionization fronts; Expansion of HII regions; Stellar Winds and Supernovas 11-Jan-2012 Stellar Content of HII Regions I Massive Stellar Evolution; Mass-Loss; Rotation; Binary interaction; Spectral features of OB stars; Runaway stars - Stellar Cluster dynamics 18-Jan-2012 Stellar Content of HII Regions II Pre--Main-Sequence (PMS) Stars; Young Stellar Systems; Stellar Initial Mass Function; Age determination & History 25-Jan-2012 Star Formation (SF) Isothermal shperes and Jeans mass; Molecular Cores collapse; Protostars 1-Feb-2012 Star Formation PMS Stellar Evolution/Contraction; Characteristics of T Tauri stars; Herbig Ae/Be Stars; Multiple SF WS 2012-2013 Lecture 7 2
Physical Processes in HII Regions Part II In this Lecture Strömgren Theory (for Hydrogen) Heating & Cooling of HII Regions The Role of Helium Forbidden Lines (CELs) Line Diagnostics for HII Regions Literature Osterbrock & Ferland, 2006, Ch. 2 Spitzer, 1978, Sec. 6.1 Tielens, 2005, Ch. 7 WS 2012-2013 Lecture 7 3 The Strömgren Theory Real HII regions are inhomogeneous. Their properties are determined by the local ionization parameter. Modeling HII regions requires a good calculation of the stellar FUV radiation field (usually through Monte Carlo). A simple (avoiding the above complications) but very useful theory to describe an HII region as a uniform spherical region, is this by Bengt Strömgren (1939). It examines the effects of the electromagnetic radiation of a single star (or a tight cluster of similar stars) of a given surface temperature and luminosity on the surrounding interstellar medium of a given density. Classical Article: Strömgren, Bengt The Physical State of Interstellar Hydrogen, The Astrophysical Journal, 89, 526-547 (1939) WS 2012-2013 Lecture 7 4
The Strömgren Theory The interstellar medium is taken to be homogeneous and consisting entirely of hydrogen. Strömgren theory describes the relationship between the luminosity and temperature of the exciting star, i.e., the intensity of the ionizing sources, on the one hand, and the density of the surrounding hydrogen gas on the other. The size of the idealized ionized region is calculated as the Strömgren radius. Strömgren s model also shows that there is a very sharp cut-off of the degree of ionization at the edge of the Strömgren sphere, because the transition region between the highly ionized and the surrounding neutral gas is very narrow, compared to the overall size of the sphere. WS 2012-2013 Lecture 7 5 The Strömgren Theory Basic Realationships The hotter and more luminous the exciting star, the larger the Strömgren sphere. The denser the surrounding hydrogen gas, the smaller the Strömgren sphere. WS 2012-2013 Lecture 7 6
The Strömgren Sphere The Strömgren sphere radius R S is determined by balancing the total rates of ionization and recombination inside it. The total ionization rate as a function of distance from the star in a spherical volume around it is: α H n H J(r) = 3S H 4πr 3 In the stationary situation, which is typical for an HII region, the number of ionizations equals the number of recombinations, and therefore: α H n H J(r) = n e 2 β 2 (T) WS 2012-2013 Lecture 7 7 The Strömgren Sphere From equating the ionization and recombinations rates we get: S H = 4π 3 R 3 Sn 2 e β 2 = 4π 3 R 3 S (nx) 2 β 2 Where S H (in S 49 10 49 photons s 1 ) is the rate at which the central star produces photons that ionize H, and x = n e /n the ionization degree. Since the gas is considered fully ionized (x ~ 1): $ R S = 3 S H ' & ) % 4π n 2 β 2 ( 1 3 WS 2012-2013 Lecture 7 8
Strömgren Spheres Characteristics $ R S = 3 S H ' 3 $ & ) S ' 61.7 & 49 ) % 4π n 2 β 2 ( % n 2 ( 1 Numerical value for T = 7,000 K. In reality n is determined by the dynamics of the HII region, i.e., its expansion into the nonuniform surrounding ISM. 1 3 pc WS 2012-2013 Lecture 7 9 Strömgren Spheres Characteristics Ionization Parameter U S and radial column density nr S. We consider a location just inside R S where x = 1, we ignore attenuation of the spectrum, and apply ionization equilibrium: U S = n π n = $ nr S = n 3 S ' & ) % 4π n 2 β 2 ( S 4πR 2 S cn = (4π /3)R 3 Sn 2 β 2 4πR 2 S cn 1/ 3 and thus, U S = β 2 3c $ = 3 ns' & ) % 4π β 2 ( $ 3 ' & ) % 4πβ 2 ( = β 2 3c nr S WS 2012-2013 Lecture 7 10 1/ 3 1/ 3 $ 3 ' = & ) % 4πβ 2 ( ( ns) 1/ 3 Both U S and nr S are proportional to (ns) 1/3. 1/ 3 ( ns) 1/ 3
Strömgren Spheres Characteristics Ionization Parameter U S and radial column density nr S. By substituting with typical numerical values, β 2 = 3.65 10 13 cm 3 s 1, n = n 2 100 cm 3 and S = S 49 10 49 photons s 1 we get: U S 3.3 10 3 ( n 2 S 49 ) 1/ 3 R S = 8.6 10 20 (n 2 S 49 ) 1/3 cm 2 The radial column density nr S is related to the average column density: N = (4π /3)nR 3 S 2 = 4 πr S 3 nr 1.15 S ( 1021 n 2 S 49 ) 1/ 3 cm 2 As n increases for fixed S, the column increases as n 1/3. Therefore, small dense HII regions can have large columns. This is the case of ultra-compact HII (UCHII) regions. WS 2012-2013 Lecture 7 11 Strömgren Spheres Characteristics The H + /H ratio. From results so far for the ionization parameter we have (Lecture 6): n H + n H 0 = U β U U H = 2 x e H α 1 c(i 4 /I 1 ) Substituting we have: U S = β 2 3c nr S n H + n H 0 = 1 3 α 1 nr S (I 4 /I 1 ) = 1 4 α 1 N(I 4 /I 1 ) which expresses the H + /H ratio in terms of the optical depth at the Lyman edge. Recalling (Lect. 6) that α 1 = 6.33 10 18 cm 2, we get: τ ν 1 = α 1 N = (6.33 10 18 )(1.15 10 21 )(S 49 n 2 ) 1/ 3 = 7280(S 49 n 2 ) 1/ 3 The H + /H is about 1/8 of this value, and thus ~900(S 49 n 2 ) 1/3. WS 2012-2013 Lecture 7 12
Strömgren Spheres Characteristics Thickness of H + /H transition region ΔR S. This is the region in which x(h) goes from 0 to 1. Its thickness is roughly the distance for an ionizing photon to be absorbed: τ ν 1 = ΔR S n H 0α 1 =1 If we neglect hardening of the spectrum and define the transition where n(h) = 0.5n, we have: ΔR S R S = 1 = 1 2 n α H 0 1 R S 1 3 8 α 1N = 2 3 U H U S I 4 I 1 3.5 10 4 (S 49 n 2 ) 1/ 3 ΔR S, as well as, H + /H ratio, U S, and nr S, all depend on the Strömgren parameter (Sn) 1/3. WS 2012-2013 Lecture 7 13 Real Strömgren Spheres The Ring Nebula (M 57) The Helix Nebula (NGC 7293) The Spirograph nebula (IC 418) All images: Hubble Space Telescope (AURA/ STScI/ NASA/ ESA) Real HII Regions are rarely circular Nonetheless, Strömgren s theory Illustrates the basic roles of photoionization and recombination. WS 2012-2013 Lecture 7 14
Ionization Balance Summary The excess energy over the ionization potential is carried away by the photo-electron as kinetic energy. Recombination is slow (~100 yr for ρ 10 3 cm 3 ), while e-e collisions occur on ~30 sec timescales. Electrons collisions exchange energy leading to Maxwell velocity distribution (Thermal emission). Thermal electrons excite low-lying levels of trace species. Downward radiative transitions cool the nebula. This energy balance sets the temperature of the gas. WS 2012-2013 Lecture 7 15 Thermal Balance Temperature of Photoionized Gas. The one important heating mechanism (photo-electric heating) involves the dissipation of the excess energy of the photoelectrons (generated by the absorption of stellar UV photons) in Coulomb collisions with ambient electrons: The mean energy of the photoelectrons is E 2 = E e = hν hν 1 ~ kt ν 1 h(ν ν 1 )α ν 4πJ ν hν dν ν 1 α ν 4πJ ν hν dν where J ν is the mean intensity of the radiation field. A detailed treatment of heating and cooling in HII regions is given in Spitzer Sec. 6.1 WS 2012-2013 Lecture 7 16
Photoelectric Heating Spitzer expresses the mean photoelectron energy in terms of the stellar effective temperature: ψ = E 2 /kt * With ζ π n H the photoionization rate per unit volume, the volumetric heating rate is Γ = ζ π n H ψkt * ψ 0 is the value near the star, ψ is averaged over an HII region. The first decreases and the second increases with T *, as does also their ratio ψ /ψ 0. WS 2012-2013 Lecture 7 17 Recombination Cooling Radiation is the main cooling mechanism of the ISM. In HII regions, radiation from recombination provides a minimum amount of cooling: each recombination drains thermal energy ½ m e υ 2 from the gas. The total cooling rate per ion is 1 2 m υ 3 α j j =k The recombination cross section α j varies as υ 2 (Lecture #6). Therefore the rate of cooling by recombination is determined by the thermal average of υ 3 υ 2 = υ, i.e., T ½. (This is confirmed by the exact calculations of Spitzer.) WS 2012-2013 Lecture 7 18
Recombination Cooling Rate The volumetric cooling, neglecting recombinations to the ground state (on-the-spot rate approximation) is Λ rec = β 2 n e n(h + )kt χ 2 φ 2 φ 2 : Recombination function (Spitzer Table 5-2, p. 107) χ 2 : Energy gain function (Spitzer Eq. 6-8, p. 135 & Table 6.2). Roughly 3/2kT of electron thermal energy is lost in each recombination in an HII region. WS 2012-2013 Lecture 7 19 Preliminary Thermal Balance for Pure H Net energy gain associated to recombinations (Spitzer Eq. 6-9) Γ ep = 2.07 10 11 n e n p T 1/2 { E 2φ 1 (hν 1 /kt) ktχ 1 (hν 1 /kt)} (energy gain resulting from captures of electrons by protons) For recombination cooling to balance with photoelectric heating requires! Γ ep = 0 T = φ T * χ ψ >1 erg cm 3 s In HII regions, where the on-the-spot approximation applies, all recaptures to the ground level can be ignored, and φ 1 and χ 1 can be replaced by φ 2 and χ 2. So, the predicted temperature is much greater than what is observed: There must be other coolants at work!! WS 2012-2013 Lecture 7 20
The Role of He in HII Regions He has high IP: He 24.6 ev (504Å); He + 54.4 ev (228Å) (see also Lecture #4) Very hot stars are needed to ionize He + (T * > 50,000 K). O-type stars are not enough, so their HII regions have no He ++. Planetary nebula stars or AGN. The radiation that ionizes He also ionizes H. He recombination radiation photoionizes H. The He threshold photoionization cross section is larger than that for H, largely compensating for its smaller abundance. WS 2012-2013 Lecture 7 21 Ionization of He by O- & B-type stars B0 star, T eff 30,000K Spectrum peaks at ~13.6 ev Many photons in 13.6-24.6 ev range Few photons with hv > 24.6 ev Two Strömgren spheres Small central He + zone surrounded by large H + region O6 star, T eff 40,000 K Spectrum peaks beyond 24.6 ev Lots of photons with hv > 24.6 ev Single Strömgren sphere H + and He + zones coincide WS 2012-2013 Lecture 7 22
Ionization Structure in Model HII Regions Osterbrock & Ferland, Astrophysics of Gaseous Nebulae and Active Galactic Nuclei, University Science Books, 2006 For an O6 star, the abundant supply of He-ionizing photons keeps both H and He ionized, whereas the smaller number generated by a B star are absorbed close to the star. WS 2012-2013 Lecture 7 23 Nebular Lines: Historical Overview Helium Discovered by Pierre Janssen in1868 in Solar emission lines (at 5816 Å), and also identified on Earth in 1895. Nebulium Discovered by William Huggins in 1864 in emission nebulae at 500.7, 495.9, and 372 nm. Identified in 1927 by Ira Sprague Bowen as [OIII] and [OII]. Significance: highlighted the possibility of long-lived quantum states and focused attention on understanding selection rules in quantum mechanics. WS 2012-2013 Lecture 7 24
Cooling of HII Regions Photoelectric heating balanced by recombination cooling in a pure hydrogen model predicted too high temperatures for HII regions. This suggests that another cooling agent is in action. Collisional excitation of elements heavier than H is a very efficient cooling process (Lecture #3). Common ions of O, N, C, Ne, Ar all have levels that are ~1eV above ground state, i.e., easily collisionally excited. Forbidden transitions due to collisional cooling from metal ions are important around T = 10 4 K. WS 2012-2013 Lecture 7 25 Cooling of HII Regions Long slit optical spectrum of the Orion Bar. The optical line emission of HII regions is dominated by the recombination lines of H & He and by the forbidden lines of heavy elements (even more so for SNRs and AGN), important for cooling. Thus, collisional excitation of heavy elements must be included in photoionization calculations. WS 2012-2013 Lecture 7 26
Optical Spectrum of an HII region Planetary Nebula NGC 3242 (ESO 1.5-m in Chile) Blue: recombination lines of H and He Red: forbidden lines of metals WS 2012-2013 Lecture 7 27 Atomic hydrogen recombination lines 1 λ = R % 1 2 n 1 ( ' 2 * & l n u ) n l : Lower level n u : Upper level R : Rydberg's constant (1.097 10 7 m 1 ) WS 2012-2013 Lecture 7 28
Collisional De-excitation The de-excitation rate coefficient, γ ul, is related to the excitation rate coefficient, γ lu, by detailed balance: γ lu = g u g l e E ul / kt γ ul A rate coefficient is a thermal average of a cross section, e.g., γ ul = σ ul (υ)υ = 4 & µ ) ( + π ' 2kT * 3 / 2 0 σ ul (υ) υ 3 e µυ 2 2kT dυ with μ the reduced mass of the system and σ ul (υ) the collisional deexcitation cross section at the relative velocity, υ, of the collision partners. The cross section and thus the rate coefficient will depend on the interaction potential of the collision partners. For, e.g., neutral partners γ ul T 1/2, while for electron ion collisions γ ul T 1/2.! WS 2012-2013 Lecture 7 29 Critical Density for collisions The two-level model (Lecture #3, Slide 16) illustrates how the cooling depends on the density of the collision partner relative to the critical density: n crit = β(τ ul )A ul β(τ ul ): escape probability of a photon formed at optical depth τ.! A ul : Einstein coefficient for spontaneous emission. γ ul : Collisional (de-excitation) rate coefficient. For HII regions, electrons are the excitation sources, and γ ul is given in standard form (Osterbrock & Ferland Eq. 3.20): γ ul = γ ul 8.629 10 6 T 1/ 2 Ω ul g u Where Ω ul is the collision strength and g u the statistical weight of level u. WS 2012-2013 Lecture 7 30
Forbidden Transitions through collisions Forbidden lines or collisionally excited lines (CELs) arise when an electron is excited by a collision into a metastable state. In high densities (~10 8 cm 3 ) the electron would almost immediately be knocked out of a metastable state by collision and not be given time to emit a photon. In low densities, the time between collisions is long enough to allow to the ion to radiate spontaneously. Typical values of γ ul are 10 7 cm 3 s 1. Osterbrock & Ferland provide tables of atomic properties of heavy elements. Table 3-15 gives a sampling of critical densities at 10,000K. For the 2p 2 ions OIII & NII: n crit (NII: 1 D 3 P; 6500 Å) = 6.6 10 4 cm 3 n crit (OIII: 1 D 3 P; 5000 Å) = 6.8 10 5 cm 3 These transitions will be sub-thermally excited in many HII regions. WS 2012-2013 Lecture 7 31 [OIII] OIII (1s 2 2s 2 2p 2 ) has two 2p electrons (isoelectronic with NII and CI). The electron spins couple to a total spin S = 0,1. The two orbital angular momenta couple to total L = 0,1,2. Of the 6 LS-coupling states, ½ satisfy the Pauli Exclusion Principle: 1 S 0 1 D 2 3 P J (J =0,1,2), with different spatial wave functions and Coulomb energies. Schematic illustration for one level of O III showing the energy level splitting for a configuration-averaged model, an L-S term split model, and a finestructure splitting model. From S. Bashkin & J. O. Stoner 1975: Atomic energy levels and Grotrian Diagrams Vols.1 & 2 WS 2012-2013 Lecture 7 32
Grotrian Diagram for the OIII triplet From S. Bashkin & J. O. Stoner 1975: Atomic energy levels and Grotrian Diagrams Vols. 1 & 2. (Labels on the solid lines refer to the transition wavelengths.) WS 2012-2013 Lecture 7 33 Forbidden Lines Cooling Transition rates for producing CELs are very low (~10 3 100 s 1 ) H recombination rates are much higher (~10 9 s 1 ) Photons are very likely to escape the nebula before being absorbed and so absorption can be ignored. They can remove a lot of heat from the nebula, resolving the high temperature issue if recombination only is considered. The higher the metallicity (i.e. heavy-element content) of a nebula, the faster it cools to thermal equilibrium, and the stronger the forbidden lines are. WS 2012-2013 Lecture 7 34
Typical Forbidden Lines (O 2+ ) (N + ) Common forbidden lines: Optical: [OIII] 4959,5007 Å, [NII] 6548,6584 Å, [SII] 6717,6731 Å Infrared: [OIII] 52,88 µm, [NIII] 57 µm WS 2012-2013 Lecture 7 35 Line Ratios as Thermometers Relative Intensities of CELs provide a measure of electron temperatures in HII regions. Widely used are the intensity ratio of the [OIII] lines λ4363/ λ5007, or λ4363/λ4959, or (λ4959+λ5007)/λ4363. Explanation: More energetic (hotter) free photoelectrons are needed to push electrons in the upper state than to populate the lower energy levels. So the line strength ratio immediately measures how hot the electron plasma in a nebula is (see previous slide). [OIII] (λ4959+λ5007)/λ4363 intensity ratio as a function of temperature. From Osterbrock (1989). WS 2012-2013 Lecture 7 36
Line Ratios as density measures Variation of [OII] (solid line) and [SII] (dashed line) intensity ratios as a function of n e at T e = 10000 K. From Osterbrock (1989). Relative Intensities of CELs provide also a measure of electron densities in HII regions. The most commonly used density measure is the intensity ratio of the [SII] lines λ6717/λ6731 or the [OII] lines λ3726/λ3729. Explanation: The de-excitation rate is only a function of electron density. WS 2012-2013 Lecture 7 37 Line Ratios as abundance measures The strengths of certain forbidden lines of heavy ions in HII regions and PNe, combined with knowledge of the electron temperature and density in the nebula, allow us to determine the abundance of these ions (and collectively of their respective element) relative to Hydrogen. For example: N(O 2+ ) N(H + ) ~ n e f (T e ) I(λ4959) I(Hβ) Where n e is the electron density; f(t e ) is the fraction of O 2+ ions able to emit at 4959 Å (with a strong dependence on nebular) and I(λ4959)/I(Hβ) is the flux of the [OII] 4959 Å line relative to Hβ. We measure the strength of the forbidden lines from all the ionic stages of an element (e.g. O, O +, O 2+ ) and add up all the abundances to find the total abundance relative to H. WS 2012-2013 Lecture 7 38
Summary Photoionization (photoelectric effect) heats a gaseous nebula. The simple Stroemgren theory describes the characteristics of a pure hydrogen nebula. Helium plays important role in real emission nebulae. Recombination & Collisional excitation cool HII regions. These cooling processes produce emission lines. These lines are used as diagnostics for characterizing HII regions. WS 2012-2013 Lecture 7 39