Chapter 2 Thermal X-ray radiation 14 2.1 Introduction Figure 2.1: The predicted X-ray spectrum of Capella. The picture illustrates the remarkable increase in sensitivity and resolution afforded by the Reflection Grating Spectrometer (RGS) of XMM-Newton by comparing the predicted RGS spectrum of the star Capella with that obtained with the EXOSAT gratings (inset). The throughput is three orders of magnitude, and the spectral resolution more than a factor of 20, better than achieved with EXOSAT. ( c G. Branduardi-Raymont, www.mssl.ucl.ac.uk/www astro/rgs/rgs impact.html) Thermal X-ray radiation is an important diagnostic tool for studying cosmic sources where high-energy processes are important. One can think of the hot corona of the Sun and of stars, solar and stellar flares, supernova remnants, cataclysmic variables, accretion disks in binary stars and around black holes (galactic and ex- tragalactic), the diffuse interstellar medium of our Galaxy or external galaxies, the outer parts of active galactic nuclei (AGN), the hot intracluster medium, the diffuse intercluster medium. In all these cases there is thermal X-ray emission or absorption. We will see in this chapter that it is possible to derive many different physical parameters from an X-ray spectrum: temperature, density, chemical abundances, plasma age, degree of ionisation, irradiating continuum, geometry etc. In this chap- ter we focus on X-ray emission and absorption in optically thin plasma’s. For op- tically thick plasmas one needs to take account of the full radiation transport in order to understand these plasmas. In that case stellar atmosphere models become 15 Figure 2.2: The observed X-ray spectrum of Capella. The spectrum is taken with the Reflection Grating Spectrometer (RGS) of XMM-Newton. The plot shows the wealth of emission lines in this source. From Audard et al., A&A, 365, L329 (2001). 16 relevant. However, we will not treat those cases here and restrict our discussion to plasmas with τ . 1. The power of high-resolution X-ray spectroscopy is shown in Fig. 2.1. This shows a simulated spectrum of Capella based on old low-resolution observations with the EXOSAT satellite. That the predictions work is shown in Fig. 2.2. This shows the success of the plasma emission codes developed at SRON by Rolf Mewe (1935–2004) and his colleagues. Figure 2.3: The observed X-ray spec- Figure 2.4: X-ray spectrum of the trum with the RGS of XMM-Newton Seyfert 2 galaxy NGC 1068. From of the flare star EQ Pegasi. The plot Kinkhabwala et al., ApJ, 575, 732 shows the so-called O vii triplet, con- (2002). sisting of the resonance, intercombi- nation and forbidden line. The rela- tive line ratio’s of this triplet contain a wealth of physical information. Image courtesy of J. Schmitt, Hamburger Sternwarte, Germany and ESA. The strength of spectral analysis lies often in the details. While there are many spectral lines, some lines contain more information than others. Fig. 2.3 shows an example of this. It is a part of the spectrum of a flare star. The three lines shown – the famous O vii triplet that we will encounter more often during this course – have an enormous diagnostic power. This triplet occurs in many other sources, with often totally different intensity ratio. See for example Fig. 2.4 and compare the intensities of the triplet with those in Fig. 2.3. While EQ Peg is in collisional ionisation equilibrium, NGC 1068 is in photoionisation equilibrium. These terms will be explained later. Up to now you only saw examples of emission spectra. A nice example of an absorption spectrum is shown in Fig. 2.5. In order to see thermal imprints on a spectrum sometimes requires the skills of a spectroscopist. See for example Fig. 2.6. There has been a fierce debate in the literature whether the sharp change in the spectrum near 18 A˚ is due to an absorption edge (lower flux to the left) or a broad emission line (higher flux to the right). Here we only note that this course is meant to train you as a spectroscopist so that you can judge such issues yourself. 17 Figure 2.5: X-ray spectrum of the Seyfert 1 galaxy IRAS 13349+2438, obtained with the RGS of XMM-Newton. From: Sako et al., A&A 365, L168. Figure 2.6: X-ray spectrum of the Seyfert 1 galaxy MCG 6-30-15, obtained with the RGS of XMM-Newton. For the proper calculation of an X-ray spectrum, one should consider three dif- ferent steps: 1. the determination of the ionisation balance 2. the determination of the emitted spectrum 3. possible absorption of the photons on their path towards Earth However, before discussing these processes we will first give a short summary of atomic structure and radiative transitions, as these are needed to understand the basic processes. 18 2.2 Summary of spectroscopic notations 2.2.1 Atoms or ions with one valence electron The electrons in an atom have orbits with discrete energy levels and quantum num- bers. The valence electron or light-electron in an atom or charged ion is the electron in the outermost atomic shell which is responsible for the emission of light. The energy configuration can be described by four quantum numbers: n - principal quantum number, characteristic for the relevant shell. It takes discrete values n =1, 2, 3, .... An atomic shell consists of all electrons with the same value of n. ℓ - quantum number of angular momentum of orbital motion of the electron: ℓ = 0, 1, 2, ... n − 1. s - quantum number of spin of electron: s = ±1/2 (for ℓ > 0) or s = 1/2 (for ℓ = 0). j - quantum number of total angular momentum of electron: j = ℓ±1/2 (for ℓ> 0) or j =1/2 (for ℓ = 0). All angular momenta are measured in units of ~ = h/2π (h is Planck’s constant). The magnitude of the angular momentum vectors is given by ~ |ℓ| = pℓ(ℓ + 1)~ ≈ ℓ~ (ℓ ≫ 1). Note that ~j = ~ℓ + ~s. For hydrogenic orbits (Coulomb field) in the theory of Bohr, the semi-major 2 axis r of an elliptical orbit in shell n is given by r = n a0/Z, where Z is the −11 nuclear charge and the Bohr radius a0 equals 5.291772108 × 10 m (∼0.529 A).˚ Note that a0 = α/4πR∞ with α the fine structure constant and R∞ the Rydberg 2 1 2 2 constant R∞ = mecα /2h; R∞hc = 2 α mec is the Rydberg energy (13.6056923 eV, 2.17987209 × 10−18 J). Thus, heavy nuclei are more compact, and also the orbits with small n are more compact than orbits with large n. The principal quantum number n corresponds to the energy In of the orbit. −2 In the classical Bohr model for the hydrogen atom the energy In = EHn with EH = 13.6 eV the Rydberg energy. Thus, electrons with large value of n have small energies (and therefore are most easily lost from an atom by collisons). Further, the characteristic velocity of the electron in the Bohr model is given by v = αZc/n. Electrons with small n or in atoms with large Z have the highest velocities. These velocities are sometimes useful to see if an interaction is important or not. For instance, if a free electron or proton collides with an atom, it will have the longest and most intense interaction with those electrons that have similar orbital velocity as the velocity of the invader. The following notation for electron orbits is frequently used: ℓ =0123456789 s p d f g h i k l m electrons and further alphabetic (with no duplication of s and p, or course!). The characters s, p, d and f originate from sharp, principal, diffuse and fundamental series (the line 19 series of alkaline spectra). The notation is complicated, but could have been worse if for example the Chinese alphabet would have been used. Example 2.1. Take the hydrogen atom. An electron in the ground state has n =1 and ℓ = 0; this is called a 1s orbit. For n = 2, ℓ = 0 we have a 2s orbit; for n = 2, ℓ = 1 we have a 2p orbit. Which are the orbits and quantum numbers for n = 3? The spin quantum number s of an electron can take values s = ±1/2, and the combined total angular momentum j has a quantum number with values between ℓ − 1/2 (for ℓ> 0) and ℓ +1/2. Subshells are subdivided according to their j-value and are designated as nℓj. Example: n = 2, ℓ = 1, j =3/2 is designated as 2p3/2. 2.2.2 Atoms or ions with more than one electron Above we summarised the situation where only one electron plays a role in the formation of spectra. Now we consider a multi-electron system. For each of the electrons one may, in analogy with the single electron configuration, give the orbital and spin angular momentum of the electrons, like ℓ~1, ℓ~2, ℓ~3,... and ~s1, ~s2, ~s3,.... Sim- ilar to the single electron system, we need to compose the total angular momentum J~ from individual angular momenta. Now one needs to consider the coupling between the electrons. In general, two different types of coupling are used, the so-called jj coupling and the Russell-Saunders or LS coupling: 1. jj coupling: j~i = ℓ~i + ~si ~ ~ X ji = J i This coupling may occur in complicated energy configurations.