Appendix 417 A. The Electromagnetic Radiation Field In this appendix, we will briefly review the most impor- The mean specific intensity Jν is defined as the tant properties of a radiation field. We thereby assume average of Iν over all angles, that the reader has encountered these quantities already 1 in a different context. Jν = dω Iν , (A.3) 4π A.1 Parameters of the Radiation Field so that, for an isotropic radiation field, Iν = Jν. The specific energy density uν is related to Jν according to The electromagnetic radiation field is described by the 4π specific intensity Iν, which is defined as follows. Con- uν = Jν (A.4) sider a surface element of area d A. The radiation energy c which passes through this area per time interval dt from where uν is the energy of the radiation field per vol- within a solid angle element dω around a direction de- ume element and frequency interval, thus measured in −3 −1 scribed by the unit vector n, with frequency in the range erg cm Hz . The total energy density of the radia- between ν and ν + dν,is tion is obtained by integrating uν over frequency. In the same way, the intensity of the radiation is obtained by dE = Iν d A cos θ dt dω dν, (A.1) integrating the specific intensity Iν over ν. where θ describes the angle between the direction n of the light and the normal vector of the surface element. Then, d A cos θ is the area projected in the direction of A.2 Radiative Transfer the infalling light. The specific intensity depends on the considered position (and, in time-dependent radiation The specific intensity of radiation in the direction of fields, on time), the direction n, and the frequency ν. propagation between source and observer is constant, With the definition (A.1), the dimension of Iν is energy as long as no emission or absorption processes are oc- per unit area, time, solid angle, and frequency, and it is curring. If s measures the length along a line-of-sight, typically measured in units of erg cm−2 s−1 ster−1 Hz−1. the above statement can be formulated as The specific intensity of a cosmic source describes its dIν = 0 . (A.5) surface brightness. ds The specific net flux Fν passing through an area el- ement is obtained by integrating the specific intensity An immediate consequence of this equation is that the over all solid angles, surface brightness of a source is independent of its distance. The observed flux of a source depends on Fν = dω Iν cos θ. (A.2) its distance, because the solid angle, under which the source is observed, decreases with the square of the −2 The flux that we receive from a cosmic source is defined distance, Fν ∝ D (see Eq. A.2). However, for light in exactly the same way, except that cosmic sources propagating through a medium, emission and absorp- usually subtend a very small solid angle on the sky. tion (or scattering of light) occurring along the path In calculating the flux we receive from them, we may over which the light travels may change the specific in- θ therefore drop the factor cos in (A.2); in this con- tensity. These effects are described by the equation of text, the specific flux is also denoted as Sν.However, radiative transfer in this Appendix (and only here!), the notation Sν will dIν be reserved for another quantity. The flux is measured =−κν Iν + jν . (A.6) in units of erg cm−2 s−1 Hz−1. If the radiation field is ds isotropic, Fν vanishes. In this case, the same amount The first term describes the absorption of radiation of radiation passes through the surface element in both and states that the radiation absorbed within a length directions. interval ds is proportional to the incident radiation. Peter Schneider, The Electromagnetic Radiation Field. In: Peter Schneider, Extragalactic Astronomy and Cosmology. pp. 417–423 (2006) DOI: 10.1007/11614371_A © Springer-Verlag Berlin Heidelberg 2006 A. The Electromagnetic Radiation Field 418 The factor of proportionality is the absorption coef- field by emission, accounted for by the τ -integral. Only −1 ficient κν, which has the unit of cm . The emission a fraction exp τν − τν of this additional energy emitted coefficient jν describes the energy that is added to the at τ reaches the point τ, the rest is absorbed. radiation field by emission processes, having a unit In the context of (A.10), we call this a formal solution of erg cm−3 s−1 Hz−1 ster−1; hence, it is the radiation for the equation of radiative transport. The reason for energy emitted per volume element, time interval, fre- this is based on the fact that both the absorption and quency interval, and solid angle. Both, κν and jν depend the emission coefficient depend on the physical state of on the nature and state (such as temperature, chemi- the matter through which radiation propagates, and in cal composition) of the medium through which light many situations this state depends on the radiation field propagates. itself. For instance, κν and jν depend on the temperature The absorption and emission coefficients both ac- of the matter, which in turn depends, by heating and count for true absorption and emission processes, as cooling processes, on the radiation field to which it is well as the scattering of radiation. Indeed, the scatter- exposed. Hence, one needs to solve a coupled system ing of a photon can be considered as an absorption that of equations in general: on the one hand the equation of is immediately followed by an emission of a photon. radiative transport, and on the other hand the equation The optical depth τν along a line-of-sight is defined of state for matter. In many situations, very complex as the integral over the absorption coefficient, problems arise from this, but we will not consider them s further in the context of this book. τν(s) = ds κν(s ), (A.7) s0 A.3 Blackbody Radiation where s0 denotes a reference point on the sightline from which the optical depth is measured. Dividing (A.6) For matter in thermal equilibrium, the source func- by κν and using the relation dτν = κν ds in order to tion Sν is solely a function of the matter temperature, introduce the optical depth as a new variable along the Sν = Bν(T ), or jν = Bν(T )κν , (A.11) light ray, the equation of radiative transfer can be written as independent of the composition of the medium (Kirch- dIν hoff’s law). We will now consider radiation propagating =−Iν + Sν , (A.8) through matter in thermal equilibrium at constant tem- dτν perature T. Since in this case Sν = Bν(T ) is constant, where the source function the solution (A.10) can be written in the form jν Sν = (A.9) Iν(τν) = Iν(0) exp (−τν) κν τν is defined as the ratio of the emission and absorption + Bν(T ) dτν exp τν − τν coefficients. In this form, the equation of radiative trans- port can be formally solved; as can easily be tested by 0 = ( ) (−τ ) + ( ) − (−τ ) . substitution, the solution is Iν 0 exp ν Bν T 1 exp ν (τ ) = ( ) (−τ ) Iν ν Iν 0 exp ν (A.12) τν From this it follows that Iν = Bν(T ) is valid for suffi- + dτν exp τν − τν Sν(τν). (A.10) ciently large optical depth τν. The radiation propagating 0 through matter which is in thermal equilibrium is de- This equation has a simple interpretation. If Iν(0) is the scribed by the function Bν(T ) if the optical depth is incident intensity, it will have decreased by absorption sufficiently large, independent of the composition of to a value Iν(0) exp (−τν) after an optical depth of τν. the matter. A specific case of this situation can be il- On the other hand, energy is added to the radiation lustrated by imagining the radiation field inside a box A.3 Blackbody Radiation 419 whose opaque walls are kept at a constant tempera- ture T. Due to the opaqueness of the walls, their optical depth is infinite, hence the radiation field within the box is given by Iν = Bν(T ). This is also valid if the volume is filled with matter, as long as the latter is in thermal equi- librium at temperature T. For these reasons, this kind of radiation field is also called blackbody radiation. The function Bν(T ) was first obtained in 1900 by Max Planck, and in his honor, it was named the Planck function; it reads ν3 ( ) = 2hP 1 , Bν T ν/ (A.13) c2 ehP kBT − 1 −27 where hP = 6.625 × 10 erg s is the Planck constant −16 −1 and kB = 1.38 × 10 erg K is the Boltzmann con- stant. The shape of the spectrum can be derived from statistical physics. Blackbody radiation is defined by Iν = Bν(T ),andthermal radiation by Sν = Bν(T ).For large optical depths, thermal radiation converges to blackbody radiation. The Planck function has its maximum at h ν 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, 10 T νmax = 5.88 × 10 Hz . (A.15) Fig. A.1. The Planck function (A.13) for different tempera- 1K tures T. The plot shows Bν(T ) as a function of frequency ν, The Planck function can also be formulated de- where high frequencies are plotted towards the left (thus large pending on wavelength λ = c/ν, such that Bλ(T ) dλ = wavelengths towards the right).