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arXiv:astro-ph/0101209v1 12 Jan 2001 1 7 0 NON-LTE RADIATION PROCESSES: APPLICATION TO THE SOLAR CORONA S. COLLIN Observatoire de Paris F–92195 Meudon, France Abstract. These lectures are intended to present a simple but relatively complete description of the theory needed to understand the formation of linesinnon-localthermodynamicalequilibrium(NLTE),withoutappealing to any previous knowledge except a few basics of physics and spectroscopy. After recalling elementary notions of radiation transfer, the chapter is fo- cussedonthecomputation ofthelevelpopulations,thesourcefunction,the ionization state, and finally the line intensity. An application is made to forbidden coronal lines which were observed during eclipses since decades. 1. Introduction Astrophysical media,at least for thefraction we can observe, areoften very dilutecomparedtothoseweareusedtoonEarth.Intheinterstellarmedium for instance, the number density is in average one atom per cm3, in HII regionsandplanetarynebulaeitisabout104 atomspercm3 andinthesolar corona 108 atoms per cm3, orders of magnitudes less than those one can get in laboratory experiments. Therefore the notion of “thermodynamical equilibrium”does nothold,and quite unusualphenomenaaretaking place, such as the emission of intense “forbidden lines” never observed on Earth. Wewillinparticularshowhowtocomputetheintensity ofvisibleforbidden coronal lines, and show how these lines, and others, can be used to get physical parameters of the solar corona. 172 Figure 1. Thespecific intensity. 2. Basics of Transfer 2.1. PHOTOMETRIC QUANTITIES We shall first define a few basic photometric quantities. Note that CGS units will generally be used. The specific intensity I : The energy dE crossing a surface of area ν ν • dA in the direction of the normal, in a solid angle dΩ, during a time dt, in a frequency interval dν, is (cf. Fig. 1): dE = I dν dΩ dt dA. (1) ν ν It can be expressed in Watt m−2 ster−1 Hz−1, but other units are used in the visible or in the X-ray range. It can be also defined per interval of wavelength, and according to the relation λν = c (speed of light), one gets I =I c/λ2. λ ν The specificintensity can beused both for the source of radiation or for the receptor, and it is constant along a path ray in the vacuum. The mean intensity J : ν • 1 J = I dΩ (2) ν ν 4π Z The flux F : ν • It is the power crossing a unit surface per unit frequency interval, in all directions (cf. Fig. 2): F = I cosθ dΩ (3) ν ν Z F is generally expressed in Jansky in the radio and far infrared range: 1 J ν = 10−23 in CGS, or 10−26 in MKS (Watts m−2 Hz−1). 173 Figure 2. The flux. Figure 3. The fluxreceived from a star. As an application, we can compute the flux from a uniform sphere ra- diating isotropically (not a very good approximation for a star, actually), cf. Fig. 3): 2π θc F = I cosθ dΩ = I dφ sinθ cosθ dθ, (4) ν ν ν Z Z0 Z0 where θ is the angle under which the sphere is seen from the observer. If c R is the radius of the star, and r its distance, one gets sinθ = R/r, and c F = πI (R/r)2. (5) ν ν Note that the flux at the surface of the sphere is πI . ν 2.2. TRANSFER EQUATION Whenalight ray does notpropagate inthevacuum,thespecificintensity is not constant: emission adds energy, and absorption removes energy. There is also diffusion, in which the global luminous energy is not changed, but can be increased in one direction, and decreased in another. We shall not take into account the polarization of the radiation, and the possibility of non stationary phenomena. 174 2.2.1. Transfer equation in a non diffusive medium The monochromatic emissivity η is defined as the power emitted per unit ν solid angle, per unit frequency interval, per unit volume. The monochromatic absorption coefficient χ is defined as follows: χ I is the power ν ν ν absorbed per unit solid angle, per unit frequency interval, by a slab of unit length normal to the direction of the propagation. Note that here χ is ν the inverse of a length. Some people use instead the “opacity coefficient”, defined per unit mass, and the emissivity per unit mass and not per unit volume. The variation of I on a path length dℓ in the direction of the light ν ray is therefore: dI = ( χ I + η )dℓ. (6) ν ν ν ν − Let us define: dτ = χ dℓ and τ = χ dℓ, (7) ν ν ν ν − − Z τ is the optical depth, which decreases towards the observer. ν The transfer equation writes then: dI ν = I S , (8) ν ν dτ − ν where S = η /χ is called the source function. ν ν ν Ifthereareseveralemissionandabsorptionprocesses,theyhavetobeall taken into account in the source function and in the absorption coefficient. For instance, if at a frequency ν a line is superposed onto a continuum, then S = Sline+Scont, and dτ = dτline+dτcont. ν ν ν ν ν ν 2.2.2. Transfer equation in a diffusive medium If there is a diffusion process, it must also be taken into account in the transferequation.Thediffusion coefficientisdefinedlike theabsorption coefficient: σ I is the power diffused per unit solid angle, per unit frequency ν ν interval, by a slab of unit length normal to the direction of the propagation. Let us assume that diffusion is coherent (i.e. without any change of frequency), and isotropic. As a consequence there is a corresponding emission coefficient, which is equal to σ J , and the transfer equation becomes: ν ν dI = [( χ σ )I + (σ J +η )]dℓ. (9) ν ν ν ν ν ν ν − − If one defines now an extinction coefficient, τtot = dτtot with dτtot = ν ν ν (χ +σ )dℓ, one gets for the transfer equation in presence of diffusion: − ν ν R dI ν = I Stot, (10) dτtot ν − ν ν 175 q t=0 Figure 4. The plane-parallel approximation. where Stot = (σ J + η )/(χ + σ ). Although it is formally similar to ν ν ν ν ν ν Eq. 8, it differs in that the intensity appears directly in the second term, so it is an integro-differential equation. Actually, the diffusion process is a probabilistic one, similar to a random walk, and one can show that the distancethataphotonwilltravelbeforebeingabsorbedisequalto[χ (χ + ν ν σ )]−1/2, while in the case of pure absorption it is equal to χ−1. ν ν In a purely diffusing medium, the transfer equation writes: dI ν = I J , (11) dτdif ν − ν ν Inthecorona,thediffusionprocessisduetoThomsonscattering byfree electrons. As a first approximation it can be considered as a coherent and isotropicprocess.Thediffusioncoefficientσ doesnotdependonfrequency, ν and is equal to σ N , whereN is the numberof electrons per unit volume, T e e and σ is the Thomsoncross section, equal to 6.65 10−25 cm2. This process T is very importantfor the continuum in the visible range, as it is responsible of the emission of the K corona. As a consequence one can consider that the corona is a purely diffusive medium for the continuum in the visible range. On the other hand we will show that diffusion is negligible in the transfer of the coronal lines, on which we will focus later on, so we will not consider it in the following sections. 2.3. APPLICATION TO A PLANE-PARALLEL MEDIUM Acommonutilisationofthetransferequationconcernsastratifiedmedium, where all physical quantities are constant on infinite parallel planes. It is the usual approximation made for stellar atmospheres (cf. Fig. 4). Often 176 interstellar clouds or diffuse nebulae are considered also as plane parallel media. The optical depth is then defined in the direction of the normal, and the transfer equation becomes, for a light ray which makes an angle θ with the normal to the planes: dI ν cosθ = I S , (12) ν ν dτ − ν The formal solution of this equation is: τ τν2 τν2 τ dτ ν ν ν I exp = S exp (13) ν ν −µ − −µ µ (cid:20) (cid:18) (cid:19)(cid:21)τν1 Zτν1 (cid:18) (cid:19) where µ = cosθ. We can apply this solution todifferent cases, according tothe boundary conditions. 2.3.1. for a stellar atmosphere In this case the boundary conditions are: no radiation incident on the surface, or I (µ < 0, τ = 0) = 0, i.e.: ν ν • τν S (t τ ν ν I (µ < 0,τ ) = exp − dt, (14) ν ν µ − µ Z0 | | (cid:18) (cid:19) the radiation at τ = remains finite, i.e.: ν • ∞ 0 S (t τ ν ν I (µ > 0,τ )= exp − dt , (15) ν ν µ − µ Zτν | | (cid:18) (cid:19) and the intensity at the surface is: ∞ τ dτ ν ν I (τ = 0) = S exp . (16) ν ν ν −µ µ Z0 (cid:18) (cid:19) One gets also the flux emerging from the surface: 1 ∞ τ dτ ν ν F (µ > 0) = 2π µdµ S exp ν ν −µ µ Z0 Z0 (cid:18) (cid:19) ∞ = 2π S (τ )dτ , (17) ν 2 ν ν E Z0 where (x)istheordernintegro-exponential: (x) = ∞exp( ux)u−ndu. En En 0 − Although Eqs. 16 and 17 are only formal solutions which require to R know the variation of S as a function of depth at all frequencies, they can ν be of some help to understand intuitively two observations. 177 First we see that the intensity is approximately equal to the source function at τ = µ, and that the internal layers below do not contribute ν to the radiation. It explains the limb darkening effect at the surface of the Sun. According to the Eddington-Barbier relation, the source function in thephotosphereisproportionaltoτ,soitdecreasestowardsthesurface(but not necessarily in the chromosphere, cf. P. Heinzel’s lectures). Thus when we observe the limb (µ 0), we are seeing layers close to the “surface” → where S is small, so I is also small. When we observe the center of the ν ν Sun (µ = 1), we are seeing deeper layers where S is large, and I is large ν ν too. So the center of the disk will appear brighter than the limb. Second,inthecaseofstars,wedonotobservetheintensity,buttheflux, which is approximately equal to the source function at τ = 1. Assuming ν again that S decreases with height in the photosphere, we can understand ν why we see absorption lines in the stellar spectrum. The absorption coefficient χ in a line is larger than in the surrounding continuum, so we are ν seeinginalinethelayersclosetosurface,andinthesurroundingcontinuum the deeper layers: the lines are then in absorption. The effect is inverted if the source function increases with the height, as it is the case for lines formed in the chromosphere, or in extended envelopes of stars: thelines are then in emission. 2.3.2. for a homogeneous slab of finite thickness One defines an optical thickness increasing towards the observer, dτ = ν χ dℓ. Since the medium is assumed homogeneous, the optical thickness of ν the slab is T = χ dℓ = χ H in the direction of the normal, where H is ν ν ν the geometrical thickness. R The boundary conditions are now: incident intensity I (τ = 0) = I , ν ν ν0 − no incident intensity at τ = T . ν ν − The solution of the transfer equation is thus (for S = const.): ν T T ν ν I (T ,µ)= I exp +S 1 exp . (18) ν ν ν0 ν − µ − − µ (cid:18) (cid:19) (cid:20) (cid:18) (cid:19)(cid:21) If T < 1, the slab is called “optically thin”; if T > 1 it is called “optically ν ν thick”. There are three interesting cases: 1. a non emissive layer (a cold cloud in front of an intense source): − T ν I (T ,µ) = I exp (19) ν ν ν0 − µ (cid:18) (cid:19) 178 2. an optically thin layer with T 1 (assuming that µ is not 1): ν − ≪ ≪ T S T ν ν ν I (T ,µ) = I exp + ν ν ν0 − µ µ (cid:18) (cid:19) T Hη ν ν = I exp + (20) ν0 − µ µ (cid:18) (cid:19) 3. an optically thick layer with T 1: ν − ≫ I (T ,µ) S . (21) ν ν ν ∼ For a finite slab with no incident radiation one can show that F = ν 2πS [0.5 (T )], and deduce that: ν 3 ν −E 1. in the optically thin case: F = 2πS T = 2πη H, or: L = 4πη ν ν ν ν ν ν − × Volume (where L is the total power emitted by the slab), ν 2. in the optically thick case: F =πS , or L = πS Surface. ν ν ν ν − × To summarize, in the optically thin case, one “sees” the emissivity, and the power is proportional to the volume. In the optically thick case, one “sees” the source function, and the power is proportional to the surface. Finally note that in a purely diffusing medium, the same equations hold, replacing S by J , and τ by the diffusion coefficient. So one gets ν ν ν in particular for an optically thin medium (i.e. Tdif 1) with no incident ν ≪ radiation on the line of sight: J Tdif I = ν ν . (22) ν µ This equation applies to the continuum of the solar corona in the visible range (the K corona). For a medium which is absorbing and diffusing, and optically thin both for diffusion and for absorption, but where the diffusion coefficient is negligible compared to the absorption coefficient, the solution of the transfer equation becomes: J Tdif +Hη I = ν ν ν . (23) ν µ This equation applies to the visible lines emitted by the solar corona (cf. later) 3. Local Thermodynamical Equilibrium (LTE) 3.1. RECALLING THE LAWS OF THERMODYNAMICAL EQUILIBRIUM (TE) The thermodynamical equilibrium is the stationary state of an ensemble of interacting particles and photons which should be achieved in an infinitely 179 thick medium (called a “Black Body”) after an infinite time. Photons and particles have then the most probable energy distribution, which corre- spondstomicroreversibility ofallprocesses.Forinstance,thereareasmany radiative (resp. collisional) excitations from the level A to the level B of an atom, as radiative (resp. collisional) deexcitations from the level B to the level A per unit time. 3.1.1. Energy distribution of photons: the Planck law It writes: 2hν3 hν −1 I B = exp 1 , (24) ν ≡ ν c2 kT − (cid:20) (cid:18) (cid:19) (cid:21) h : Planck constant = 6.6262 10−27 erg s, − k : Boltzmann constant = 1.3806 10−16 erg K−1 − T : temperature. − Caution: it can also beexpressed in units of wavelength, and it writes then: 2hc2 hc −1 B = exp 1 , (25) λ λ5 λkT − (cid:20) (cid:18) (cid:19) (cid:21) which has different shape and position of the maximum. The integration over ν or λ gives: σT4 B = B dν = , (26) ν π Z where σ is the Stefan constant = 5.6696 10−5 erg cm−2 s−1 K−4. There are two limiting cases : for hν kT, the Rayleigh-Jeans law (used in radio-astronomy): − ≪ 2hν3 kT 2kT B = = ; (27) ν c2 hν λ2 for hν kT, the Wien law (used in the X-ray range): − ≫ 2hν3 hν B = exp . (28) ν c2 −kT (cid:18) (cid:19) 3.1.2. Energy distribution of particles in non quantified levels: the Maxwell law It gives the number of particles per unit volume whose velocity projected on an axis z is between v and v +dv : z z z M 1/2 Mv2 dN = N exp z dv , (29) z z (cid:18)2πkT(cid:19) −2kT !