The electromagnetic radiation field A In this appendix, we will briefly review the most important The flux is measured in units of erg cm2 s1 Hz1.Ifthe properties of a radiation field. We thereby assume that the radiation field is isotropic, F vanishes. In this case, the same reader has encountered these quantities already in a different amount of radiation passes through the surface element in context. both directions. The mean specific intensity J is defined as the average of I over all angles, A.1 Parameters of the radiation field Z 1 J D d!I ; (A.3) The electromagnetic radiation field is described by the spe- 4 cific intensity I, which is defined as follows. Consider a D surface element of area dA. The radiation energy which so that, for an isotropic radiation field, I J.Thespecific passes through this area per time interval dt from within a energy density u is related to J according to solid angle element d! around a direction described by the n 4 unit vector , with frequency in the range between and u D J ; (A.4) C d,is c where u is the energy of the radiation field per D dE I dA cos dt d! d; (A.1) volume element and frequency interval, thus measured in erg cm3 Hz1. The total energy density of the radiation is where describes the angle between the direction n of the obtained by integrating u over frequency. In the same way, light and the normal vector of the surface element. Then, the intensity of the radiation is obtained by integrating the dA cos is the area projected in the direction of the infalling specific intensity I over . light. The specific intensity depends on the considered posi- tion (and, in time-dependent radiation fields, on time), the direction n, and the frequency . With the definition (A.1), A.2 Radiative transfer the dimension of I is energy per unit area, time, solid angle, and frequency, and it is typically measured in units of The specific intensity of radiation in the direction of propa- 2 1 1 1 erg cm s ster Hz . The specific intensity of a cosmic gation between source and observer is constant, as long as no source describes its surface brightness. emission or absorption processes are occurring. If s measures The specific net flux F passing through an area element the length along a line-of-sight, the above statement can be is obtained by integrating the specific intensity over all solid formulated as angles, dI Z D 0: (A.5) ds F D d!I cos : (A.2) An immediate consequence of this equation is that the sur- The flux that we receive from a cosmic source is defined in face brightness of a source is independent of its distance. The exactly the same way, except that cosmic sources usually observed flux of a source depends on its distance, because the subtend a very small solid angle on the sky. In calculating solid angle, under which the source is observed, decreases 2 the flux we receive from them, we may therefore drop the with the square of the distance, F / D [see (A.2)]. factor cos in (A.2); in this context, the specific flux is However, for light propagating through a medium, emission also denoted as S. However, in this Appendix (and only and absorption (or scattering of light) occurring along the here!), the notation S will be reserved for another quantity. path over which the light travels may change the specific P. Schneider, Extragalactic Astronomy and Cosmology, DOI 10.1007/978-3-642-54083-7, 583 © Springer-Verlag Berlin Heidelberg 2015 584 A The electromagnetic radiation field intensity. These effects are described by the equation of the other hand, energy is added to the radiation field by 0 radiative transfer emission, accounted for by the -integral. Only a fraction 0 0 exp of this additional energy emitted at reaches dI the point , the rest is absorbed. D I C j : (A.6) ds In the context of (A.10), we call this a formal solution for the equation of radiative transport. The reason for this is The first term describes the absorption of radiation and states based on the fact that both the absorption and the emission that the radiation absorbed within a length interval ds is coefficient depend on the physical state of the matter through proportional to the incident radiation. The factor of propor- which radiation propagates, and in many situations this state tionality is the absorption coefficient , which has the unit of depends on the radiation field itself. For instance, and cm1.Theemission coefficient j describes the energy that j depend on the temperature of the matter, which in turn is added to the radiation field by emission processes, having depends, by heating and cooling processes, on the radiation a unit of erg cm3 s1 Hz1 ster1; hence, it is the radiation field it is exposed to. Hence, one needs to solve a coupled energy emitted per volume element, time interval, frequency system of equations in general: on the one hand the equation interval, and solid angle. Both, and j depend on the of radiative transport, and on the other hand the equation of nature and state (such as temperature, chemical composition) state for matter. In many situations, very complex problems of the medium through which light propagates. arise from this, but we will not consider them further in the The absorption and emission coefficients both account context of this book. for true absorption and emission processes, as well as the scattering of radiation. Indeed, the scattering of a photon can be considered as an absorption that is immediately followed by an emission of a photon. A.3 Blackbody radiation The optical depth along a line-of-sight is defined as the integral over the absorption coefficient, For matter in thermal equilibrium, the source function S is Z solely a function of the matter temperature, s 0 0 .s/ D ds .s /; (A.7) S D B.T / ; or j D B.T / ; (A.11) s0 where s0 denotes a reference point on the sightline from independent of the composition of the medium (Kirchhoff’s which the optical depth is measured. Dividing (A.6)by law). We will now consider radiation propagating through and using the relation d D ds in order to introduce matter in thermal equilibrium at constant temperature T . the optical depth as a new variable along the light ray, the Since in this case S D B .T / is constant, the solution equation of radiative transfer can be written as (A.10) can be written in the form dI D DI C S ; (A.8) I./ I.0/ exp . / d Z C 0 0 B .T / d exp (A.12) where the source function 0 D C j I.0/ exp . / B.T / Œ1 exp . / : S D (A.9) From this it follows that I D B.T / is valid for sufficiently is defined as the ratio of the emission and absorption coeffi- large optical depth . The radiation propagating through cients. In this form, the equation of radiative transport can be matter which is in thermal equilibrium is described by the formally solved; as can easily be tested by substitution, the function B .T / if the optical depth is sufficiently large, solution is independent of the composition of the matter. A specific case of this situation can be illustrated by imagining the I./ D I.0/ exp ./ radiation field inside a box whose opaque walls are kept at a Z constant temperature T . Due to the opaqueness of the walls, C 0 0 0 d exp S./: (A.10) their optical depth is infinite, hence the radiation field within 0 the box is given by I D B.T /. This is also valid if the This equation has a simple interpretation. If I.0/ is the volume is filled with matter, as long as the latter is in thermal incident intensity, it will have decreased by absorption to equilibrium at temperature T . For these reasons, this kind of avalueI.0/ exp ./ after an optical depth of .On radiation field is also called blackbody radiation. A.3 Blackbody radiation 585 The function B.T / was first obtained in 1900 by Max Planck, and in his honor, it was named the Planck function; it reads 3 2hP 1 B.T / D ; (A.13) c2 ehP=kBT 1 27 where hP D 6:625 10 erg s is the Planck constant 16 1 and kB D 1:38 10 erg K is the Boltzmann constant. The shape of the spectrum (Fig. A.1) can be derived from statistical physics. Blackbody radiation is defined by I D B.T /,andthermal radiation by S D B .T /.Forlarge optical depths in the case of thermal radiation, the specific intensity converges to blackbody radiation. For small optical depth, the radiation field is approximated by an integral over the emissivity j, which can deviate strongly from that of blackbody spectrum even in the case of a thermal source; an example is the optically thin bremsstrahlung from the hot gas in galaxy clusters (see Sect. 6.4). The Planck function has its maximum at h P max 2:82 ; (A.14) kBT i.e., the frequency of the maximum is proportional to the temperature. This property is called Wien’s law. This law can also be written in more convenient units, T D 5:88 1010 Hz : (A.15) max 1K The Planck function can also be formulated depending on wavelength D c=, such that B.T / d D B.T / d, Fig.