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Introduction to Astrophysical Spectra Solar Spectropolarimetry: from Virtual to Real Observations 2019 Introduction to Astrophysical Spectra Solar spectropolarimetry: From virtual to real observations 2019 Jorrit Leenaarts ([email protected]) Jaime de la Cruz Rodr´ıguez([email protected]) 1 Contents Preface 4 1 Introduction 5 2 Solid angle 6 3 Radiation, intensity and flux 7 3.1 Basic properties of radiation . 7 3.2 Definition of intensity . 7 3.3 Additional radiation quantities . 9 3.4 Flux from a uniformly bright sphere . 11 4 Radiative transfer 12 4.1 Definitions . 12 4.2 The radiative transfer equation . 14 4.3 Formal solution of the transport equation . 16 4.4 Radiation from a homogeneous medium . 17 4.5 Radiation from an optically thick medium . 19 5 Radiation in thermodynamic equilibrium 22 5.1 Black-body radiation . 22 5.2 Transfer equation and source function in thermodynamic equi- librium . 24 6 Matter in thermodynamic equilibrium 25 7 Matter/radiation interaction 26 7.1 Bound-bound transitions . 27 7.2 Einstein relations . 29 7.3 Emissivity and extinction coefficient . 31 7.4 Source function . 33 8 Line broadening 33 8.1 Broadening processes . 34 8.2 Combining broadening mechanisms . 36 8.3 Line widths as a diagnostic . 40 2 9 Bound-free transitions 40 10 Other radiative processes 43 10.1 Elastic scattering . 45 10.2 The H− ion ............................ 46 11 Rate equations 46 11.1 General theory . 46 11.2 The two-level atom . 47 11.3 A three-level atom as a density diagnostic . 49 11.4 A three-level atom as a temperature diagnostic . 52 12 Exercises 53 3 Preface These are a fraction of the lecture notes for the Solarnet Summer School - Solar spectropolarimetry: From virtual to real observations. These notes are extracted from the AS5004 - Astrophysical Spectra course as given at Stockholm University. The latter are very much a work in progress and draw from many different sources. We want to give explicit credit to the most important ones here: • Handwritten lecture notes by Alexis Brandecker • The lecture notes “Radiative transfer in stellar atmospheres” by Robert J. Rutten • The lecture notes “The generation and transport of radiation” by Robert J. Rutten • The book “Radiative processes in astrophysics” by George B. Rybicki and Alan P. Lightman. • The book “Astrophysics of Gaseous Nebula and Active Galactic Nuclei” by Donald E. Osterbrock As these notes are a work in progress they will be continually updated during the course. Please send me an email if you find (or suspect) any errors. Jaime de la Cruz Rodr´ıguez& Jorrit Leenaarts 4 1 Introduction Electromagnetic radiation (from now on just radiation or photons) has been and will remain the primary medium from which we obtain information from astrophysical objects. Cosmic rays – which consist of protons, nuclei of heav- ier elements and electrons – and neutrinos are highly interesting other sources of information. They are however emitted by a much more restricted range of sources. Thus they cannot provide as wide a view of the Universe as radiation. Photons do not decay, so they can carry information over literally cosmic distances as long as they are not absorbed by intervening matter. We can measure the direction, energy distribution as a function of wavelength (the spectrum) and direction of oscillation (the polarization state) of radiation and the time variation of these quantities, which all can be used to infer properties of astrophysical objects. Radiation is important for another reason: it influences the physical struc- ture of many objects. It is for example an energy transport mechanism in stars, stellar winds can be driven by radiation pressure, starlight can heat the interstellar medium around them, and radiation was the primary contributor to the cosmic energy density during the radiation-dominated era in the early Universe. Knowledge of the formation and transport of electromagnetic radiation is thus essential for an astrophysicist. In these notes we will have a closer look at a macroscopic description of radiation and its interaction with matter, with the main quantity being called intensity. We will also look at the detailed processes on the scale of atoms and molecules that can absorb and emit photons. We will then develop equations that couple the macroscopic description of radiation to the microscopic processes, so that we can describe the expected radiation given the properties of the emitting material, and conversely, given observed radiation, infer properties of the source. 5 10 CHAPTER 2. BASIC RADIATIVE TRANSFER By defining Iν per infinitesimally small time interval, area, band width and solid an- gle, Iν represents the macroscopic counterpart to specifying the energy carried by a bunch of identical photons along a single “ray”. Since photons are the basic carrier of electro- magnetic interactions, intensity is the basic macroscopic quantity to use in formulating radiative transfer1.Inparticular,thedefinitionpersteradianensuresthattheintensity along a ray in vacuum does not diminish with travel distance — photons do not decay spontaneously. z r sin θ ∆ϕ r ∆θ θ y ϕ x Figure 2.1: Solid angle in polar coordinates. The area of the sphere with radius r limited by (θ, θ + ∆θ) Figureand (ϕ 1:, ϕ +Solid∆ϕ)is angler2 sin θ in∆θ spherical∆φ so that ∆Ω coordinates.=sinθ ∆θ ∆ϕ. The area of the sphere with radius r limited by (θ; θ+∆θ) and (φ, φ+∆φ) is r2 sin θ ∆θ ∆φ so that ∆Ω = sin θ ∆θ ∆φ. Adapted from Rutten (2003). Mean intensity. The mean intensity Jν averaged over all directions is: 1 1 2π π 2 Solid angleJ (⃗r,t) I dΩ = I sin θ dθ dϕ. (2.2) ν ≡ 4π ν 4π ν ! !0 !0 2 1 1 1 In orderUnits: to erg properly cm− s− describeHz− ster− radiation,,justasfor weIν need.Inaxialsymmetrywiththe to introduce the conceptz-axis of (θ 0) along the axis of symmetry (vertical stratification only, “plane parallel layers”) J solid angle.≡ This is best done in analogy with the concept of angle. Theν simplifies to, using dΩ =2π sin θ dθ = 2π dµ with µ cos θ: angular size of an object in a 2D plane− is seen≡ from the origin is the length of the projection of the object1 π on the unit circle.1 +1 This is of course how the Jν (z)= Iν(z,θ)2π sin θ dθ = Iν (z,µ)dµ. (2.3) 4π 0 2 1 radian is defined, and thus! the maximum angular!− size of an object is: This quantity is the one to use when only the availability of photons is of interest, irrespec- tive of the photon origin, for exampleZ 2 whenπ evaluating the amount of radiative excitation and ionization. dφ = 2π: (2.1) 0 Flux. The monochromatic flux is: Similarly we can define the solidFν angle as the surface area of the projection 2π π of an object on the(⃗r, unit⃗n, t) sphereI cos centredθ dΩ = on theI cos origin.θ sin θ d So,θ dϕ. an object(2.4) that Fν ≡ ν ν completely encloses the origin! has a solid!0 angle!0 of Units: erg s 1 cm 2 Hz 1 or W m 2 Hz 1.Thisisthenetflowofenergypersecond − − − 2π − π − through an area placed at locationZ ⃗r Zperpendicular to ⃗n .Itisthequantitytouseforspec- ifying the energetics of radiationΩ = transfer,sin throughθ dθ dφ stellar= 4π: interiors, stellar atmospheres,(2.2) 0 0 1Except for polarimetry, which needs three more Stokes parameters discussed in Section 6.1 on page 137. An infinitesimal solid angle dΩ can be interpreted as an infinitesimal surface on the unit sphere. In spherical coordinates: dΩ = sin θ dθ dφ (2.3) We will use solid angle in the definition of intensity. 6 ∆Ω ^ r l ∆Ω ∆Ω ∆Α Figure 2: Definition of intensity. Photons fly through the surface with area ∆A located at ~r, in a solid angle ∆Ω around the direction ^l. For simplicity the normal n^ of ∆A is taken parallel to ^l in this figure. The energy carried by these photons is proportional to ∆A and ∆Ω if they are chosen so small that the radiation field is constant across these intervals. 3 Radiation, intensity and flux 3.1 Basic properties of radiation Radiation can be described as waves with different frequencies or as a stream of photons with different energies. In the wave description the frequency ν, wavelength λ and the speed of light in vacuum c are related as c ν = : (3.1) λ The energy of a photon associated with a frequency ν is given by hc E = hν = ; (3.2) λ where h is Planck’s constant. One can express photon energy or wave fre- quency with a formal temperature as well: E hν hc T = = = ; (3.3) k k λk where k is Boltzmann’s constant. In astronomy wavelengths are often expressed in units of Angstr¨om(with˚ symbol A),˚ which is 10−10 m = 0:1 nm. 3.2 Definition of intensity The fundamental quantity with which we shall describe radiation is the in- tensity Iν. It is defined so that for an area dA with normal n^ and located 7 at ~r, the amount of energy transported through this surface between times t and t + dt in the frequency band (ν; ν + dν) in a cone with solid angle dΩ around direction ^l is ^ ^ dE = Iν(~r; l; t) (^n · l) dA dt dν dΩ: (3.4) The intensity can be expressed in various units, commonly used units are: −2 −1 −1 [Iν] = W m Hz ster (3.5) −1 −2 −1 −1 [Iν] = erg s cm Hz ster (3.6) −2 −1 −1 [Iλ] = W m m ster (3.7) In Eq. 3.7 the intensity is defined for a wavelength band instead of a fre- quency band.
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