Download Notes on Photoionized Regions 1. Introduction 2. Hydrogen Nebulae

Notes on Photoionized Regions Wednesday, January 12, 2011 CONTENTS: 1. Introduction 2. Hydrogen Nebulae A. Ionization equations B. Recombination coefficients and cross sections C. Structure of the hydrogen nebula 3. Hydrogen-­Helium Nebulae A. He+ zones B. He2+ zones 4. Metal ionization A. Fundamental processes B. Specific ions 1. Introduction We are now ready to consider the basics of photoionized regions. The major cases to be considered are:  H II regions powered by young stars;  Planetary nebulae powered by dying stars; and  Active galactic nuclei, where a central engine (the accretion disk around a black hole) ionizes surrounding gas clouds. All of these have in common a source of ionizing photons (generally extreme UV, or photons with energies >13.6 eV). The surrounding gas is ionized and heated by these photons, and emits radiation both via recombinations and due to collisional excitation of metals (and also by assorted minor mechanisms). The structure and appearance of these regions will be the subject of the next 3 lectures. For the purpose of this first lecture, we will calculate only the ionization structure. We will assume the gas to have a uniform density and temperature, which will be fixed in our calculation. In the next lecture we will investigate what determines the temperature of an ionized nebula, and later in the course (when we discuss hydrodynamics) we will discuss the density structure of the ISM. The material for this section can be found in Chapter 2 of Osterbrock & Ferland. 2. Hydrogen Nebulae We first consider the idealized situation of a pure hydrogen nebula. Since hydrogen is the most abundant element (comprising ~92% of the atoms in the Universe) this is often a good first approximation. A. IONIZATION EQUATIONS We begin by considering the illumination of hydrogen gas by a central radiation source. We will define: Gas parameters: n(H) = nH = total density of all hydrogen nuclei (atomic, ionized, or molecular) [cm−3] ne = electron density [cm−3] np = proton density [cm−3] ξ = ionization fraction (=np/nH for pure hydrogen) Radiation parameters: Lν = source luminosity [erg s−1 Hz−1] Jν = mean radiation intensity [erg cm−2 s−1 Hz−1 sr−1] Q(H0) = rate of production of photons capable of ionizing H0 [s−1] Physics parameters: aν = photoionization cross section [cm2] α(H0,T) = rate coefficient for recombination, H + + e− → H 0 + γ [cm3 s−1] The rate of generation of H0-­‐ionizing photons is simply an integral over the source luminosity, €
∞
L
Q(H 0 ) = ∫ν (H 0 ) ν dν , i
hν
where the cutoff frequency is the ionization energy of hydrogen (13.6 eV or 3.29 PHz). This is of order 1048−1050 photons s−1 for an O star. €
In a spherically symmetric nebula with density nH, one may then write the equation for the ionization rate. The electron and proton densities are both nHξ. Thus: ∞
4 πJν (r)
n Hξ˙ (r) = −α (H 0 ,T)n H2ξ 2 (r) + n H[1− ξ (r)] ∫ν (H 0 )
aν dν , i
hν
where the first term accounts for recombinations and the second for ionizations. In order to go further, we will approximate the radiation field Jν to be entirely from € star. It is attenuated by the inverse-­‐square law and by absorption as one moves the central outward, r
L
4 πJν (r) = ν 2 exp − ∫ 0 n H[1− ξ (r')]aν dr' , 4 πr
[
€
]
where the integral in the exponential represents the optical depth for absorption of extreme UV photons.1 In this formula we have neglected changes in the structure of the nebula on the timescale for the photons to propagate through it (almost always valid) and radiation emitted within the nebula (generally wrong at the tens of percents level – more on this later). A common further approximation is to neglect the frequency dependence of the cross section. This is not good in general – the photoionization cross section decreases with frequency – but since for stellar sources most of the radiation is emitted near 13.6 eV one can get an approximate solution this way and gain conceptual insight. This leads to r
Q(H 0 )a
ξ˙ (r) = −α (H 0 ,T)n Hξ 2 (r) +
[1− ξ (r)]exp − ∫ 0 n H [1− ξ (r')]a dr' , 2
4 πr
which is the basic equation that we will examine. €
B. RECOMBINATION COEFFICIENTS AND CROSS SECTIONS We next need to know the actual values of the recombination coefficient and the photoionization cross section. Recombination: Recombination is simply due to the radiative decay of an electron from an unbound to a bound state of hydrogen. Therefore, we might imagine that the rate of the reaction (in units of recombinations cm−3 s−1) is simply the probability to find a proton and electron “near” each other times the atomic decay rates. Here “near” means within a few Bohr radii (i.e. a volume of V~10−23 cm3) and so the probability is P~nenpV. The typical atomic decay rate is A~109 s−1, so we would expect a recombination coefficient of α~10−14 cm3 s−1. In fact, at T=104 K, the recombination coefficient for hydrogen is: α A (H 0 ,10 4 K) = 4.2 ×10−13 cm3 s−1 . The larger value than our naïve estimate is due to Coulomb focusing of the incoming electron. This focusing becomes more important at lower temperatures, so roughly €
αA∝T−0.7. Note that we have appended the subscript A. In general, a hydrogen atom can recombine either directly to the ground state, H + + e− → H 0 (1s) + γ , or indirectly to an excited state (n≥2). In the latter case the hydrogen atom reaches the ground state by emitting spectral lines (and possibly 2s1s continuum). The €
1 The repeated factor of 4π is not a typo: the 4πJ comes from the definition of J being per ν
ν
steradian of photon propagation direction, whereas the 4π on the right-­‐hand side is associated with the surface area of a sphere of radius r surrounding the star. [
]
recombination coefficient αA includes both of these reaction channels. However, the reaction where the H atom goes directly to the ground state emits a photon with hν>13.6 eV. This photon can therefore ionize an H atom. In most nebulae, therefore, recombinations directly to the ground state do not result in a net recombination. Thus it may be a better approximation to exclude them. If we do so, we arrive at a different recombination coefficient, α B (H 0 ,10 4 K) = 2.6 ×10−13 cm3 s−1 . The two situations – where photons above 13.6 eV are re-­‐absorbed or not – are called Case A recombination and Case B recombination respectively, and their reaction rates are € B) recombination coefficients. called Case A (or Case Photoionization: Photoionization is simply the inverse process of (radiative) recombination. Therefore, it can be computed using the principle of detailed balance, which states that in thermal equilibrium, the forward and reverse rate of every reaction cancel. If we imagine a gas in thermal equilibrium at temperature T (with kT<<13.6 eV), the rate of recombinations directly to the ground state (in cm−3 s−1) is α1s(T)nenp. By comparison, the rate of photoionizations out of the ground state is: ∞
4 πJν (r)
n(H 0 ) ∫ν (H 0 )
aν dν . i
hν
The radiation density for a blackbody is however given by Wien’s law, €
2hν 3
Jν = 2 e−hν / kT , c
so the principle of detailed balance states: €
∞
2ν 2
α1s (T)n e n p = n(H 0 ) ∫ν (H 0 ) 2 e−hν / kT aν dν . i
c
This can be compared to the Saha equation, €
n e n p ⎛ 2πm e kT ⎞ 3 / 2 −hν i (H 0 )/ kT
= ⎜
, ⎟ e
n(H 0 ) ⎝ h 2 ⎠
to give: €
⎛ h 2 ⎞ 3 / 2 ∞ 2ν 2 −h[ν −ν (H 0 )]/ kT
i
α1s (T) = ⎜
aν dν . ⎟ ∫ν (H 0 ) 2 e
i
c
⎝ 2πm e kT ⎠
€
This allows us to obtain aν from the recombination coefficient (the two form a Laplace transform pair). The value at threshold is 6×10−18 cm2. C. STRUCTURE OF THE HYDROGEN NEBULA We are now ready to consider a typical H II region associated with star formation. The typical timescale associated with its ionization evolution is (αBnH)−1; for nH ~ 10 cm−3, this is ~104 yr, which is short compared to the lifetime of an O star. Therefore, we expect the ionization structure to have time to reach equilibrium, and we set ξ(r,t)=ξ(r). Then: r
Q(H 0 )a
0
2
0 = −α (H ,T)n Hξ (r) +
[1− ξ (r)]exp − ∫ 0 n H[1− ξ (r')]a dr' . 2
4 πr
It is also convenient to define the radius r1 within which the rate of recombinations at full ionization would equal the rate of emission of source photons, €
4 3
πr1 α Bn H2 = Q(H 0 ) . 3
If we then let u = r/r1, we have €
u
C
ξ 2 (u) = 2 [1− ξ (r)]exp −C ∫ 0 [1− ξ (u')] du' , 3u
where we have defined the dimensionless parameter C = nHr1a. For our typical H II region, with a photon emission rate of 1049/s, we find r1 = 4.5×1019 cm = 15 pc and C = 3000. € case is that C >> 1. This case can be solved by writing the above Indeed, the typical equation in the form. u
1 d
ξ 2 (u) = − 2
exp −C ∫ 0 [1− ξ (u')] du' . 3u du
In the deep interior of the nebula, where the left-­‐hand side is approximately 1, this admits the solution: €
[
]
[
]
[
[
u
]
]
exp −C ∫ 0 [1− ξ (u')] du' = 1− u 3 , or 3u 2
. C(1− u 3 )
€
Thus for u<1, the interior of the nebula is almost fully ionized. At u~1 the above solution breaks down; there is a transition region of thickness Δu~C−1/3 over which the nebula becomes mostly neutral. B€
eyond this, there is an exponentially decreasing flux of photons ξ (u) = 1−
and the outside material is neutral. Thus, an ionized bubble with a sharp edge is formed. This bubble is called a Strömgren sphere. Its radius is r1. 3. Hydrogen-­Helium Nebulae We are now ready to add helium in to the mix. We note here that the first ionization energy of helium (He0He+) is 24.6 eV, whereas the second ionization energy is 54.4 eV. The He:H ratio (by number of atoms) is 0.08 for primordial gas, although the modern-­‐day values tend to be somewhat larger. A. He+ ZONES We now consider a source that emits some number of photons capable of ionizing helium, ∞
L
Q(He 0 ) = ∫ν (He0 ) ν dν . i
hν
For cool sources, Q(He0) is usually quite a bit smaller than Q(H0), although this will not be true for planetary nebulae or AGN. In this case, a helium Strömgren sphere appears inside the hydrogen sphere. Its €
radius is given by 4 3 2
πr2 n H f He (1+ f He )α B (He 0 ,T) = Q(He 0 ), 3
where fHe is the He:H ratio by number, αB(He0,T) is the Case B recombination coefficient for He+He0 (i.e. to all excited states of He), and we have used the fact that the densities of He+ and electrons in t€
he He+ zone are nHfHe and nH(1+fHe), respectively. There should be a correction to the above equations because some of the helium-­‐
ionizing photons will get absorbed by the small amount of neutral hydrogen present within the He+ zone. But usually this is a small correction. At 24.6 eV, the cross sections for photoionization of He0 and H0 are 8✕10−18 cm2 and 1.1×10−18 cm2, respectively, so any photon capable of ionizing helium usually does (exceptions below). The existence of the helium ionization zone also modifies the hydrogen ionization zone. This is for two reasons. One is that photons above 24.6 eV are absorbed by helium and so are not available to ionize hydrogen. However, when the ionized helium recombines, it radiates helium spectral lines and two-­‐photon continuum radiation, some of which emerges at hν>13.6 eV. The probability p of emitting such a photon can be computed by taking the recombination coefficients to each excited level and following their decay chains with the appropriate branching probabilities, giving p = 0.96. (In high-­‐density regions, n > few ✕ 103 cm−3, the metastable 1s2s 3 S1e level can collide with an electron and transition to 1s2s 1Se0 . In the case of this decay path, we find p = 0.66.) In this case, if we compute the total number of hydrogen recombinations and set this equal to the number of ionizations, we get €
€
4
4
π (r13 − r23 )n H2α B (H 0 ,T) + πr23 n H2 (1+ f He )α B (He 0 ,T) = Q(H 0 ) − (1− p)Q(He 0 ) . 3
3
If we approximate p≈1, then the usual hydrogen Strömgren sphere radius is unchanged. €
The fraction of the volume of the ionized nebula in which helium is also ionized is then: −1
⎡ Q(H 0 )
r23
α B (He 0 ,T)
α B (He 0 ,T) ⎤
=
−1+ p −
⎢
⎥ . r13 α B (H 0 ,T) f He (1+ f He ) ⎣Q(He 0 )
α B (H 0 ,T)(1+ f He ) ⎦
In the limit p1, fHe<<1, and the two recombination coefficients being equal (typically good to ~10%, since Case B recombination coefficients only weakly distinguish between €
recombination on a proton versus an He+), this is r23
Q(He 0 )
≈
. r13 f HeQ(H 0 )
For harder spectra, in which the Q(He0):Q(H0) ratio is larger, more of the helium in the nebula is ionized, as we expect. For very hard sources, €roughly a central star temperature exceeding ~4✕104 K (O7), we have Q(He0):Q(H0)>fHe. In this case, the He+ zone fills the entire ionized nebula. The He+ zone cannot extend outside of the H+ zone, since H0 absorbs the >24.6 eV photons, so in these cases a single ionization zone develops with a boundary determined by the flux of >24.6 eV photons. B. He2+ ZONES We now consider nebulae with central sources emitting photons above 54.4 eV. These could include planetary nebulae, AGN, and the postulated supermassive metal-­‐free stars (“Population III”) that may have existed in the first galaxies. As one might guess, this leads to the formation of a He2+ zone surrounding the central source, whose radius is given by 4 3 2
πr3 n H f He (1+ 2 f He )α B (He + ,T) = Q(He + ). 3
We have r3<r1 even for source temperatures T*~105 K. €
4. Metal Ionization Metal atoms are rare in nature: for example, solar abundances2 are (by number) O:H = (4.9±0.6)×10−4, C:H = (2.7±0.3)×10−4, and even lower for everything else. Therefore as far as the ionization structure is concerned, they have no major effect and merely respond to their environment. (As far as the thermal structure and emitted spectra are concerned they will be very important.) Their ionization state in each zone is determined by their ionization potential, photoionizations and recombinations, and charge-­‐transfer reactions. A. FUNDAMENTAL PROCESSES Metal atoms can recombine in two ways. One is the standard radiative recombination to any state of the atom, e.g.: O 2+ (2p 2 3 P e ) + e− → O + (2p 2 4 So ) + γ . In some cases, the rate can be enhanced by dielectronic recombination, which is the inverse of autoionization followed by radiative decay, e.g.: €
C 2+ (2s2 1Se ) + e− ↔ C + (2s2p( 3 P o )3d 2 F o ),
C + (2s2p( 3 P o )3d 2 F o ) → C + (2s2p 2 2 De ) + γ , C + (2s2p 2 2 De ) → C + (2s2 2p 2 P o ) + γ .
Note that the initial ion is unstable against autoionization, but sometimes decays radiatively. €
Photoionization can proceed either by direct photoionization, or excitation to an autoionizing state. Note that photoionization depends on the availability of photons above the ionization energy of the atom in question. There is a cutoff energy in the spectrum of such photons of 13.6 eV (in an H I region), 24.6 eV (in an H+/He0 zone), 54.4 eV (in a He+ zone), or ∞ (in a He2+ zone). Finally, charge transfer, in which an ion collides with a neutral atom and transfers an electron, can be important: O + + H ↔ O + H + . When such reactions are exothermic, they have zero activation energy (due to the induced dipole attraction) and do not require a photon to be radiated. Thus their rate coefficients €
δ(T) tend to be of order 10−9 cm3 s−1, i.e. in neutral regions they are much faster than recombinations, and in ionized regions they can be of comparable speed. B. SPECIFIC IONS We now discuss the ionization state of the major ions. The cases of O and C are illustrative. 2 Asplund, Grevesse, Sauval & Scott, ARA&A 47:481—522 (2009). Oxygen – The first ionization energy of oxygen is 13.6 eV, i.e. almost the same as for H. This results in oxygen rapidly neutralizing due to charge transfer in H I regions. The second ionization energy is 35.1 eV, so in He0 regions oxygen cannot be ionized above the O II stage. In He+ regions, however, oxygen can be photoionized to O III, with the O II/O III ratio determined in part by how many photons the source emits above 35.1 eV. Ionization to the next stages (O IV and O V) requires 54.9 or 77.4 eV, and thus only occurs in He2+ zones around sources with very hard spectra. Ionization to O VI (114 eV) is generally not possible except with X-­‐ray sources or collisions in shock-­‐heated gas. Carbon – Carbon has an ionization energy of only 11.3 eV, so it can be ionized to C II by radiation that leaks outside of the main photoionized nebula. For this reason, carbon ionization is usually the main source of electrons in “neutral” gas. The second ionization energy is 24.4 eV, so ionization to C III requires the helium to be at least singly ionized (at practical temperatures it can charge-­‐transfer with He0). The third ionization energy is 47.9 eV, so some C IV could be present in the He+ zone for extremely hard sources. Finally the fourth ionization energy is 64.5 eV, which is only accessible within the He2+ zone, and for very hard sources. Further ionizations involve the inner shell and are only accessible via X-­‐
rays or extremely hot gas. Zone: H I H+/He0 He+ He2+ Hydrogen H I H II H II H II Helium He I He I He II He III Carbon C II C II C III/C IV C IV/C V Nitrogen N I N II N III N IV/N V Oxygen O I O II O II/O III O IV/O V Neon Ne I Ne II Ne II/Ne III Ne III/Ne IV Magnesium Mg II Mg III Mg III Mg III/Mg IV Silicon Si II Si III Si III—V Si V Sulfur S II S II S III—V S V/S VI Iron Fe II Fe III Fe III/Fe IV Fe V/Fe VI You should be aware that:  Some of the metals – particularly Fe – are often locked up in dust grains and in such cases their atomic lines will be weak or absent.  Which spectral lines are observed will depend not only on the presence of particular species but also whether they have spectral lines in the waveband of observation and whether such lines can be excited. For example, in the neutral regions surrounding an ionized nebula, the electron density and temperature are usually low, so the optical lines of the indicated species will not be seen. Also, the noble gas, np1, and np5 ions do not have low-­‐lying optical lines.