THE INTERSTELLAR MEDIUM (ISM) Summary Notes: Part 2

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1 THE INTERSTELLAR MEDIUM (ISM) Summary Notes: Part 2 Dr. Mark WESTMOQUETTE The main reference source for this section of the course chapter 5 in the Dyson and Williams (The Physics of the Interstellar Medium) book. Beware: this book is very technical!! Note: Figure numbers refer either to the chapter and figure number (N-n) in Freedman and Kauffman, or to the Dyson and Williams textbook. 2 PHOTOIONIZED NEBULAE: H ii REGIONS H ii regions (like the Orion nebula) are regions of ISM gas in which an OB star (or OB stars) are embedded. For simplicity we shall start by assuming an idealized nebula which is spherical and at its centre is a single O or B star. The radiation from the central OB star PHOTOIONISES the nebula gas, and it is this process and its inverse (RECOMBI- NATION) which provides the physical basis for much of the nebula s physical properties. 2.1 The photoionisation process Photons from the central OB star, which has a high effective temperature ( K at B0 and up to K for an O5 star) can have a sufficiently high energy such that when they are absorbed by atoms in the gas they cause the removal of a bound electron from the atom. This is termed IONISATION. We will consider Hydrogen since it is the most abundant element in the gas: our basic picture of the atomic structure of H is that it consists of a proton and a single electron, which can occupy any bound energy state, denoted by a quantum number n = 1,2,3,4, etc. The n = 1 state is known as the GROUND STATE and has zero energy (E = 0; see Fig 5-25). As we go to very large n-values the energy levels become very close together and there comes a point where the electron is no longer in a bound state but can occupy a range of possible CONTINUUM states where it has a certain free kinetic energy. The removal of a bound electron from an atom to form an ion+electron is called ION- IZATION. If the energy supplied to ionise the atom comes from an incoming photon, the process is called PHOTOIONISATION. Using Hydrogen as an example (Fig 5-25) we see that the energy required to take the 1

2 Table 1: Hydrogen recombination lines (wavelengths in nm) Lyman Balmer Paschen n n = 1 n n = 2 n n = 3 α β γ δ ɛ 94 electron from the ground state (n = 1, E = 0) to the continuum is 13.6 ev (electron volts). This value is known as the ionisation energy for Hydrogen, I H. Thus to ionise H from the ground state we need photons with energy >13.6 ev, or in wavelength terms photons with wavelengths LESS than 912 Å. Only hot (OB) stars have radiation fields with significant numbers of photons at these short (UV) wavelengths. Thus only OB stars can photoionise the hydrogen gas in the surrounding nebulae. NB. the low density of the gas in H ii regions means that the EXCITATION of the H-atoms is very low, so that nearly ALL the H-atoms will have their electrons in the ground state (n = 1), so photoionisation need ONLY be considered from the n = 1 state. The strong (Coulomb) attraction between the negatively charged electron (e ) and the positively charged proton (p + ) produces the inverse process of RECOMBINATION, so we have a REVERSIBLE reaction: H + hν p + + e (1) In recombination the energy of the photon which is produced depends on two things: (a) the Kinetic Energy (KE) of the recombining electron (in its continuum state) and (b) the binding energy of the bound-level into which the recombination occurs. If the recombination occur directly to some upper level n, the electron will try and go to the ground state by jumping down to lower n-levels, and each n n jump produces a photon at a specific energy or wavelength which corresponds to some particular spectral line in Hydrogen. For transitions from n n = 2 give rise to the Balmer series of lines seen at optical wavelengths (Hα is the 3 2 transition and occurs at 6563 Å, Hβ is the 4 2 transition and occurs at 4861 Å). Transitions from n n = 3 give the Paschen series in the near IR, whilst those from n n = 1 give the Lyman series in the far-uv (see Table 2.1). This is the main process by which the Hydrogen emission lines are produced in H ii region spectra and this produces the RECOMBINATION EMISSION LINE spectrum of the nebula. Equilibrium occurs when there is a balance between the forward and backward rates of the above equation this is called the CONDITION OF IONIZATION BALANCE. Photon energy is described using the formula: E = h ν, where h is the Planck constant and ν is the frequency of the light (photon). This can be related to the wavelength using c = ν λ, where λ is the wavelength, thus E = h c / λ. In this way we can relate energy in ev to wavelength. 2

3 2.1.1 Thermalisation of gas in a pure Hydrogen nebula As eqn. 1 shows, the nebula gas contains three types of particles: neutral atoms (H), protons (p + ) and electrons (e ). Photoionisation feeds energy continuously into the nebula via the kinetic energy (KE) of the ejected (photo)electrons. The energy of the electrons is determined by the energy distribution of the ionising photons from the ionising OB star. This is determined by its effective temperature, T eff. Collisions between the particles take place which THERMALIZES the particle velocity distributions and transfers energy from one type of particle species to another, and this occurs on such short timescales as to be regarded as instantaneous compared to the timescale for successive photionisation/recombination. After thermalization, all the particles in the photoionised nebula have a Maxwellian Distribution of velocity (and hence KE), and thus we can assign a single KINETIC temperature, T K, of the gas particles which is the same for all three types of particle. The speeds at which the particles interact and share energy follows the order: (i) hot, photoelectrons share their energy with other electrons, (ii) electrons transfer energy to protons via collisions (needs a longer time since the proton mass is > electron mass), (iii) protons share their energy by collisions with neutral atoms Ionisation balance in (pure H) H ii regions We need to be able to calculate the forward and background rates to get the Hydrogen ionisation ratio. The PHOTOIONISATION rate is given by: N PI = a n HI J [m 3 s 1 ] (2) n HI is the number density of neutral H (H i) J is the number of ionising photons crossing unit area per unit time at some point in the nebula, a is the photoionisation cross-section for H (from level n = 1). To a first approximation we can assume a = m 2. 3

4 The CROSS-SECTION of an interaction can be thought of as the effective area that the interacting particle sees. The term cross-section is just a way to express the PROBA- BILITY (likelihood) of interaction. See Fig Info Figure 1: Interaction cross-section. Geometric vs. effective area. The RECOMBINATION rate into level n of an H-atom is: N R = n e n p α n = (n e ) 2 α n (T e ) [m 3 s 1 ] (3) n e is the electron density n p is the proton density α n = RECOMBINATION COEFFICIENT of level n, and is the RATE of ion electron recombination per unit volume (units cm 3 s 1 ). It varies as a function of T e (the electron temperature of the gas). Note: here the interaction cross-section is already incorporated in α n. To get the TOTAL rate of recombination we need to sum eqn. 3 over all levels of H (n = 0 ) Photoionisation is assumed to ONLY occur from level n = 1 (due to the low excitation of the gas) BUT recombination can occur to any n-value level. Thus any recombination directly to the n = 1 level is IMMEDIATELY followed by a photoionisation. For this reason, astronomers make it easy for themselves when calculating equation 3, and ONLY take recombinations to level n = 2 and above. This is known as CASE B RECOMBINATION. Under the assumption of case B, α n becomes α B, and a good approximation is: α B = Te 3/4 Calculations often use the DEGREE of IONIZATION, X, of H which is given by the simple relation: n p = n e = X n where n is the number density of hydrogen nuclei (i.e. the sum of the proton and neutral atom number densities). Thus n HI = n Xn, and note that X lies between 0 and 1. n e n p = (n e ) 2 because in equilibrium, the number of electrons equals the number of protons 4

5 Figure 2: Ionisation structure of two homogeneous H + He model H ii regions. The condition of IONIZATION EQUILIBRIUM says that eqn. 2 = eqn. 3, i.e. the total photoionisation rate = total recombination rate. This allows us to calculate X for any given value of J and for a specified gas density (and hence n e ) at some distance r from the star. If the star is emitting S ionising photons per second (i.e. below 912 Å), then J = S /(4πr 2 ) [m 2 s 1 ] If we take a typical O-star energy distribution for an O6V star (like those of the Trapezium stars in Orion), we find S photons per sec, and adopting for example a gas density of n = 10 8 m 3 and r = 1 parsec, gives J m 2 s 1. This gives X = , i.e. the H gas is almost completely ionised. This can be done for a range of r-values (distances from the central star) to give X as a function of r see Fig Sizes of photoionised H ii regions: STRÖMGREN SPHERES A particular star cannot ionise an indefinite volume of ISM gas. The volume it can ionise depends on the volume at which the total recombination rate is exactly equal to the rate at which the star emits ionising photons (S). 5

6 If this region has a radius R S, the condition of ionisation balance says that: S = 4π 3 R3 S n 2 α B For a given value of S and n (the nebula gas density) we can solve the equation easily to find R S, which is called the STRÖMGREN RADIUS for the ionised gas around an OB star. Typical values are: S = photons per sec, n = 10 8 m 3, which gives R S = 3 parsecs. The radius of a Strömgren Sphere (the region of fully ionised H gas around an OB star) will INCREASE if the gas density is LOWER, and will also INCREASE the HOTTER is the central OB star since it will emit more ionising photons. For low densities (say n = 10 6 ) and for an O6V star we can get R S 100 parsecs. 2.3 Heating and cooling processes in photoionised nebulae Photoionisation feeds energy into the nebula gas since it creates hot, fast photoelectrons. Without some ENERGY LOSS mechanisms the gas temperature would rise indefinitely. Observational analysis of H ii region spectra suggest electron temperatures of about K (= 10 4 K), significantly lower that the effective temperatures of the ionising OB stars. If we have a pure H nebula, energy is released in H recombination. The KE of the recombining electron is converted to photons, which may escape from the nebula and cool it. However, by comparing the heating and cooling rates from this process we find that higher nebula temperatures are expected. On average each recombination removes an energy (3/2) k T e from the gas, and the energy loss, or COOLING RATE, is: L = 3 2 k T e N R [Joules m 3 s 1 ] The energy input, the HEATING RATE, is given by: G = N I Q [Joules m 3 s 1 ] where Q is the energy injected into the gas in each photoionisation. For star with an approximate Blackbody spectrum, one can show that Q (3/2) k T eff. In equilibrium, L = G, and N R = N I, and we expect to find T e T eff, i.e. a nebula gas electron temperature comparable to the star s effective temperature. For a typical OB star T eff K, which is much higher that values of T e K for nebulae derived from their spectra. Inclusion of other possible H-cooling process like Balmer-line emission, and Hydrogen freefree emission (which occurs mainly at radio frequencies) still gives too high temperatures, This implies that we have neglected some important cooling process... 6

7 2.3.1 Forbidden line cooling of H ii regions (and PNe) Real nebulae contain other elements than just Hydrogen. We must, therefore, relax our assumption of a pure H nebula. A special emission line formation process in metals can give emission line radiation which can escape from the gas and cool it. This is called FORBIDDEN LINE emission. Bound electrons in atoms can be excited/de-excited in two ways: by photons (resulting in absorption/emission) or by collisions (non-radiative process, meaning there is no way to see it) between atoms or atoms and electrons. The Balmer (n 2) and Lyman (n 1) transitions in Hydrogen are excited/de-excited by the absorption/emission of photons, and give rise to what are called ALLOWED transitions in quantum mechanics. The downward (de-excitation) transitions of electrons within the atom occur spontaneously with a very high probability (typical transition probabilities are 10 9 per sec). However, some energy levels in metals (and their ions) are split into sub-levels or fine structure states. Downward transitions amongst these low-lying energy levels can have a very low probability of occurring spontaneously (typically per sec or 1 per secs). Because of this they are often also called METASTABLE states. Photons emitted in this way result in what are called FORBIDDEN LINES. The formation process of these Forbidden Lines provides the cooling of the nebula gas in the following way: (a) In the ISM, the atom density is too low for atom atom collisions to occur. However atom electron collisions do occur. Bound electrons in the lower levels of e.g. O + and O ++ are excited to higher levels by COLLISIONS with free electrons in the gas REMOVING some of the Kinetic Energy of the colliding electrons. These free electrons come from the photoionisation of H. Excitation potentials of these fine structure levels are 1 ev, which is approximately equal to the energy of the electrons (kt 1 ev at T e = K). Note: all levels of H and He have much higher excitation potentials, meaning they cannot be excited by collisions with electrons of the same temperature. (b) Fine structure states are metastable, so the electron sits there for a relatively long time ( secs). Remember, collisions with atoms are negligible. This gives the electron the chance to RADIATIVELY de-excite by itself (spontaneously), thus emitting a forbiddenline photon. The energy is now transferred to the photon. Note: if the gas density is too high, the jump to the lower level can be induced by another collision before spontaneous de-excitation has time to occur. This is called collisional deexcitation no photon is emitted but kinetic energy is returned to the gas. For most H ii regions the rate of radiative (spontaneous) de-excitation is greater that collisional de-excitation. Every forbidden transition has a critical density, above which the collisional de-excitation (by electron atom collision) rate exceeds the radiative (spontaneous) de-excitation rate. In these circumstances we say the forbidden line becomes quenched. 7

8 Figure 3: Energy level diagram for left: O ii (O + ) and right: O iii (O ++ ). Despite the low abundance of metals (compared to H), forbidden transitions make a significant contribution because their excitation potentials (1 few ev) are so low. Only electrons with thermal (energy kt ) energies are needed, rather than ionising photons (E > 13.6 ev). (c) because the photon emitted in the Forbidden Line transition has only a very weak probability of being absorbed the emitted photon easily escapes from the nebula gas, carrying away photon energy and this cools the gas. By this process we have REMOVED kinetic energy from the gas and transformed this to photon energy which ESCAPES from the gas easily. When this process is included in calculations of the heating and cooling rates it turns out that irrespective of the T eff of the central OB star, we end up with electron temperatures of between K, as observed. The term forbidden is really a misnomer! It was coined before we understood all the rules governing electron transitions. A more intuitive name for forbidden lines is collisionallyexcited lines (CELs), since they are emitted following collisions of free electrons with ions in the nebular plasma. For OXYGEN, the low energy states of singly ionised Oxygen (O + ) and doubly ionised Oxygen (O ++ ) are illustrated in Fig. 3. Forbidden Transitions amongst these levels include the lines at 3726, 3729 Å for O + and 5007, 4959, 4363 Å for O ++. Note: the ionisation potential energy of neutral O is about the same as for H (13.5 ev), whilst that of O + is about 35.1 ev. Thus only the hottest O stars can produce significant O ++ in their H ii regions. Not all CELs result from forbidden transitions it just happens that most optically observable CELs are forbidden. Moving only slightly into the UV non-forbidden CELs can start to be seen (e.g. C iv λλ1548, 1551, or Si iv λλ1394, 1403). + Info 8

9 Figure 4: Energy level diagram for N ii (N + ) and O iii (O ++ ), showing the most commonly used temperature diagnostic ratios. The arrow indicates increasing energy. 2.4 How to derive temperatures, densities and the chemical composition of emission-line nebulae (H ii regions, PNe) It turns out that the observed relative intensities of forbidden lines in species like O +, O ++ and other forbidden lines in elements like Sulphur, Argon, Chlorine provide good ways of measuring the electron temperatures and electron densities in H ii regions. The most widely used nebular thermometer is the intensity ratio of the [O iii] lines λ4363/λ5007, or equivalently λ4363/λ4959 [or even (λ4959+λ5007)/λ4363] see Figs 4 and 5. Intuitively, the temperature dependence can be seen from the fact that collisions with more energetic (hotter!) free photoelectrons are needed to push the bound electrons of the O ++ ion in the upper state from where 4363 Å line is emitted than to populate the lower energy levels from where 4959 or 5007 Å are emitted. So the line strength ratio λ4363/λ5007 immediately tells us how hot the electron plasma in a nebula is. In a cool nebula the λ4363/λ5007 ratio will be lower than in a hotter nebula. Similarly, the most commonly used density measure is the intensity ratio of the [S ii] lines λ6717/λ6731 or the [O ii] lines λ3726/λ3729. Fig. 6 shows the energy level diagrams for these ions, where the transitions used are highlighted. Fig. 7 plots the density dependence of these line ratios for a given nebula temperature, T e. It is immediately obvious that the usefulness of these indicators only spans a finite range in densities. The strengths of certain forbidden lines of heavy ions in an H ii region or a Planetary Nebula, 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. The abundance of O ++, for example, can be calculated using the following formula: N(O 2+ ) N(H + ) n e I(λ4959) f(t e ) I(Hβ) 9

10 Figure 5: [O iii](λ4959+λ5007)/λ4363 intensity ratio as a function of temperature. Reproduced from Osterbrock (1989). Figure 6: Energy level diagram for O ii (O + ) and S ii (S + ), showing the most commonly used density diagnostic ratios. 10

11 Figure 7: Variation of [O ii] (solid line) and [S ii] (dashed line) intensity ratios as a function of n e at T e = K. Reproduced from Osterbrock (1989). where n e is the electron density; f(t e ) is the fraction of O 2+ ions able to emit at 4959 Å (this has a strong dependence on nebular temperature see Fig. 8) and I(λ4959)/I(Hβ) is the flux of the [O ii] 4959 Å line relative to Hβ. Now we can 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 Hydrogen. The technique is as follows: we first observe the nebula using a spectrograph coupled to a telescope. We measure the line strengths ( fluxes ) relative to the strength of a representative Hydrogen recombination line. The one typically used is Hβ 4861 Å (since it s strong and easily accessible in the optical part of the spectrum). We use the observed ratios of the strengths of certain forbidden lines to determine the temperature and density of the nebular plasma. We then use expressions such as that shown above to find out the abundance of ions relative to H +, and add up the abundances of all the ionic stages to find the total abundance of a heavy element (e.g. C, N, O, Ne, S, Ar, Cl, etc.) relative to Hydrogen in the nebula. Only with large telescopes can the weaker lines of the above elements be observed from nebulae in our Galaxy and beyond, allowing us to study the composition of distant galaxies. This information is vital for a proper understanding of the chemical evolution of the universe and the recycling of matter within galaxies (from stars to the ISM and vice-versa). 11

12 Figure 8: Fraction of O ++ temperature. ions able to emit the 4959 Å line as a function of nebula 2.5 Additional information UV forbidden lines (e.g. C iii] 1908 Å; [O iii] 1663 Å) are sampled from space (e.g. from the IUE or FUSE satellites). For IR lines (e.g. [O iii] 52, 88 µm; [Ne iii] 57 µm) we need a high altitude plane flying above most of water vapour in the atmosphere, since water absorbs in the IR (e.g. the Kuiper Airborne Observatory KAO) or a satellite (e.g. Spitzer). For optical lines (e.g. [O iii] 4363, 4959, 5007 Å, [O ii] 3727, 3729 Å, [N ii] 6548, 6584 Å, [Ar iii] 7005 Å, [Ar iv] 4740 Å, [Ne iii] 3968 Å, etc.), we can use ground based telescopes. Ionized metal atoms can also emit recombination lines (allowed/permitted transitions), but are usually very weak (due to their low abundances). Like H recombination lines, they exhibit little or no dependence on nebular temperature or density so are not very useful. HELIUM, which makes up about 10% (by number) of the gas in nebulae, does not emit forbidden-lines, but recombination lines just like Hydrogen does. The most prominent of these, and most accessible, are in the optical part of the spectrum: He i 4471 Å, 5876 Å, 6678 Å, and He ii 4686 Å. Doubly ionised Helium however can only be found in Planetary Nebulae, as the cooler stars of H ii regions do not emit photons energetic enough to ionise He +. Hence the He ii 4686 Å line (produced when He ++ recombines with an electron to produce He + ) is NOT observed in H ii regions. + Info The abundance of Helium in nebulae is derived from the observed strengths of the aforementioned Helium recombination lines relative to Hβ. Sometimes He i recombination lines in the radio part of the spectrum are also used. Because of the way that Helium and Hydrogen recombination lines are created, the strengths of these lines do not depend very sensitively on the electron temperature in a nebula. Thus, even without a very accurate measurement of the nebular temperature, a rather accurate Helium abundance (relative to H) can be derived. 12