Radiation Processes Black Body Radiation Heino Falcke Radboud Universiteit Nijmegen Contents: Planck Spectrum Kirchoff & Stefan-Boltzmann Rayleigh-Jeans & Wien Einstein Coefficients Literature: Based heavily on Radiative Processes in Astrophysics, G.B. Rybicki & A.P. Lightman, John Wiley & Sons, New York (Chap. 1)
Thermal Radiation Black Body Radiation Every body at a certain temperatures radiates. If it wouldn t, thermodynamics wouldn t work. Assume you have a hot body in a cold photon environment. The energy will flow from the hot (energetic) body to the cold (less energetic) photon gas - the body emits energy. If the photon gas is more energetic, the photons will tend to heat the body - the body absorbs energy. This will continue until absorption and emission are in equilibrium. On a more practical level, you could picture temperature as Brownian motion of particles which move randomly. If an external low-energy photon is scattered off the walls of such a microscopically vibrating body, the photons will get a kick (see Inverse Compton scattering later) and more energetic photons will come back from the wall. If the radiation field is very strong, photons would start to give the particles a kick, thereby loosing energy and heating up the wall. We now consider a closed box in thermal equilibrium at a single temperature T with radiation intensity I ν inside. We find the following properties:
photon number is not conserved - photons can be created and destroyed at the walls, thermal equilibrium (i.e. T) will adjust the number of photons I ν is independent of properties of enclosure (except T). Assume joining two enclosures at the same temperature which are separated by a narrow filter. If I ν I ν energy would flow despite thermal equilibrium (contradicts thermodynamics). I ν must be isotropic and unpolarized for the same reasons. B ν is called the Planck function. I ν = universal function of T and ν = B ν (T) Kirchhoff s Law for Thermal Emission As discussed before, the intensity of the source will converge towards the source function, hence S ν = I ν = B ν (T) and from the definition of the source function S ν = j ν /α ν, we have. j ν = α ν B ν (T) The fact that the black body radiation is a universal function depending only on Temperature and frequency is called Kirchhoff s Law (but doesn t yet give a functional form).
The Planck Spectrum The functional form of the black body radiation was eventually derived by Planck, first through interpolating the asymptotic Rayleigh-Jeans and Wien laws, and then by statistical arguments using quantized oscillators to emit and absorb radiation: the beginning of quantum physics. The basic idea is that the thermal energy leads to the excitation of a range of oscillators (waves), however, there is only a finite number of oscillators and states possible. Each oscillator is characterized by a wave number k = 2π/λ = 2πν/c = 2π/tc (2πR/tc = v/c = kr = 1 1/k is the radius of circle a charge has to run around with c to produce this wave). In a finite box of length l, there are finite standing waves with multiples of the wavelength possible, hence the maximum number of states is given by the number of wavelengths across the box n = l/ λ = l k 2π In three dimensions we have the total number of states as Since we can set l 3 = V one gets N = ( n) 3 = l3 d 3 k (2π) 3 N V = d3 k (2π) 3
This is the number of modes or states of the photon per volume. However, there is another degree of freedom, since the waves can be in two polarizations. Hence we actually have: N V = 2 d3 k (2π) 3 Now, we express the wavenumber infinitesimal in spherical coordinates, such that with k 2 = (2πν/c) 2 and dk = (2π/c)dν we get d 3 k = k 2 dkdω d 3 k = (2πν/c) 2 (2π/c)dνdΩ = (2π/c) 3 ν 2 dνdω going back to the equation above, we can replace d 3 k and find N V = ν 2 dνdω 2(2π/c)3 (2π) 3 N V dνdω = ν 2 2(2π/c)3 (2π) 3 = 2 ν2 c 3 Note: That this could also be written as 2/λ 3 ν, so that the infinitesimally small box size volume per frequency, where only one state is distinguishable, has a box length of order the wavelength.
Now that we know how many states there are in a phase cell, we need to know how much energy there is per state. We consider a quantum phase cell, i.e. one state, for a specific frequency ν. The state may contain n photons of total energy E n = nhν. The probability of a state of energy E n is proportional to e E n/k B T (Maxwell). Hence the average energy is: < E >= n=0 E n e E n/k B T n=0 e E n/k B T with some transformations one can find that < E >= hνe hν/k BT 1 e = hν hν/k BT e hν/k BT 1 This equation resembles the expression for Bose-Einstein statistics with a limitless number of particles with chemical potential = 0. To get the energy per solid angle per volume per frequency, we have to multiply the average energy with the number of states, given by ρ s = 2ν2 c 3 U = E ρ s and I = U c This is an energy density and hence to get to intensity we have to multiply by c (U = I/c, see above).
B ν (T) = 2hν3 /c 2 e hν/k BT 1 This has cgs units of erg s 1 cm 2 ster 1 Hz 1. Expressed per unit wavelength we get (because of ν = c/λ and dν = cdλ/λ 2 ) B λ (T) = 2hc2 /λ 5 e hc/λk BT 1
Rayleigh-Jeans Law The low frequency end (which you will most often encounter): hν k B T Expand the exponential Hence we find e hν k BT 1 = hν k B T +... I ν (T) = 2ν2 c 2 k BT The high frequency end: hν k B T Wien Law e hν k BT 1 B ν (T) = 2hν 3 /c 2 e hν/k BT This means we have a very fast exponential decay beyond the maximum. This often causes numerical problems when students try to calculate black bodies on the computer...
Properties of Planck Law Monotonicity with Temperature A Planck curves with a higher temperature lies at all frequencies above the Planck curve of a lower temperature. Wien Displacement Law Question: where does the Planck law have its maximum? B ν ν = 0 ν max = 2.82144 k BT h = 5.87893 1010 Hz T K The peak frequency of the Black Body spectrum scales linearly with the temperature. Similarly one can calculate the maximum wavelength and finds: λ max = 0.29 cm K T The maximum wavelength does not correspond to the maximum frequency, i.e. λ max c/ν max.
Stefan-Boltzmann Law We can also integrate the Planck spectrum over all frequencies and solid angles, to get the total energy radiated per time and surface area. First we calculate the flux coming off one side of a uniformly and isotropically radiating disk along the disk axis. F ν = 2π 0 ( π/2 0 cosθ sinθdθ = 1/2) dφ π/2 0 cosθb ν dω = 2πI ν π/2 0 B ν cosθ sinθdθ = πb ν Next, we integrate the (solid angle integrated) Planck spectrum over frequency: This is typically written as: F = 0 πb ν dν = π 2k B 4 π 4 T 4 15c 2 h 3 F = σt 4, which is the Stefan-Boltzmann law, where the Stefan-Boltzmann constant σ is defined as σ = 2k B 4 π 5 erg = 5.6704 15c 2 10 5 h3 s cm 2 K 4. This means that the flux radiated per surface area of a black body is a very strong function of the temperature (4th power). Hence, hotter bodies cool much faster and hotter stars (e.g., O/B stars) are many orders of magnitude brighter that the sun even though the temperature only changes by one order of magnitude.
Brightness temperature Especially in radio astronomy one often expresses the brightness of a source in terms of a brightness temperature. The brightness temperature of a source is the temperature a black body of the observed size would need to have in order to reproduce the flux of the observed flux. Hence, for any observed frequency ν obs we define T B, such that I νobs = B νobs (T B ) Note, that for a non-blackbody spectrum T B changes as a function of ν obs. At low frequencies, where the Rayleigh-Jeans law is applicable, one has I ν = 2ν2 c k BT 2 B T B = c2 I 2ν 2 ν k B This is often used in radio astronomy, since one can express the flux of a source easily. For a source with a Gaussian brightness distribution (as often fitted in high-resolution VLBI images) over an angular size θ, Condon et al. (1982) give a simple formula T B = 1.2 10 9 K(S ν / mjy )(θ/ mas ) 2 (ν/ GHz) 2 Indeed many radio sources have such high brightness temperature. From this one can derive that the emission process is non-thermal. Since, such a high temperature would predict very strong
emission at very high frequencies (X-rays) that is not seen. However, if the source becomes optically thick, the brightness temperature approaches the temperature of the source (this can be shown by expressing the radiation transfer equation in term of T B and using the Rayleigh-Jeans approximation for its solution). Effective Temperature Similar to the brightness temperature for the single frequency flux one can also define an effective temperature for the total integrated flux. The effective temperature is the temperature a black body would need to have in order to radiate as much energy integrated over all frequencies as the observed source of the same size radiates. F = cos θi ν dνdω = σt 4 eff T eff = (F/σ) 1/4
Einstein Coefficients Definition Kirchhoff s law j ν = α ν B ν relates emission to absorption for a thermal emitter. This must have some equivalent relation on the microscopic level between absorption and emission by matter (Atoms)! Einstein discovered this relationship, by considering two discrete energy levels 1 & 2, given by energy E which is populated with statistical weight g 1 and E + hν with statistical weight g 2 ( statistical weight, here relates to a degeneracy factor, i.e. there are multiple states with essentially the same energy). Transition of 2 1 means emission, and transition of 1 2 means absorption of photons with energy hν.
Einstein defined three processes: Processes 1. Spontaneous Emission: Can occur any time whether a radiation field is present or not. The transition probability per unit time is given by A 21 (in units of sec 1 ). A 21 is called the Einstein A-coefficient. 2. Absorption: Can occur when a photon of energy hν is available. The transition probability per unit time will be proportional to the number of available photons (i.e. the intensity) and a factor B 12. One can then define the transition probability per unit time as B 12 J ν, where J ν = 1 Iν dω 4π is the mean intensity of the radiation density. One little complication is that the frequency of the transition is not completely sharp, but has a line profile. The shape is given by a function φ(ν) (see Figure) which is normalized such that 0 φ(ν)dν = 1. Therefore the transition probability [sec 1 ] is actually (B 12 is called the Einstein B-coefficient) B 12 Jν with Jν = J 0 ν φ(ν)dν
3. Stimulated Emission: This is the weirdest and an absolutely non-intuitive process. The emission of a photon (decay) is stimulated by the presence of another photon of the same frequency. This is also proportional to the intensity of the radiation: B 21 J Somehow Einstein needed that process to explain the Planck function, but how did he get that idea... (OK, he was a genius and I am not). Thermal Equilibrium In thermodynamic equilibrium, we have that the number of transitions from 1 2 equals the number of transitions from 2 1 (i.e. nothing changes): n 1 B 12 J = n2 A 21 + n 2 B 21 J if we have n 1 and n 2 being the number densities of atoms in level 1 and 2, respectively. n 1 B 12 J n2 B 21 J = (n1 B 12 n 2 B 21 ) J = n 2 A 21 J = J = A 21 (n 1 /n 2 )B 12 B 21 = A 21 /B 21 (n 1 /n 2 )(B 12 /B 21 ) 1 n 2 A 21 (n 1 B 12 n 2 B 21 ) We have solved for the average intensity and expressed everything on the right in terms of ratios.
In thermodynamic equilibrium, the population density is given by the Maxwell-Boltzmann distribution 1 and here we have: n 1 g 1 e E/k BT = n 2 g 2 e = g 1 e hν/k BT (E+hν)/(k BT) g 2 so we can put this into the equation above J = ( g1 B 12 g 2 B 21 A 21 /B 21 ) e hν/k BT 1 Now comes the clever twist: in thermodynamic equilibrium, we know what the intensity has to be. It is blackbody radiation described by the Planck function, hence we set J ν = B ν. If B ν is more or less linear over the narrow frequency range ν, then we can set J = J ν. So, we have: B ν = 2hν3 /c 2 e hν/k BT 1 = A 21 /B 21 ( ) g1 B 12 e hν/k BT 1 g 2 B 21 g 1 B 12 = g 2 B 21 & A 21 = 2hν3 c 2 B 21 1 Wikipedia says: The Maxwell-Boltzmann distribution can be derived using statistical mechanics. It corresponds to the most probable speed distribution in a collisionally-dominated system consisting of a large number of non-interacting particles in which quantum effects are negligible. Since interactions between the molecules in a gas are generally quite small, the Maxwell-Boltzmann distribution provides a very good approximation of the conditions in a gas.
We have a connection between the different microscopic Einstein coefficients. If we can determine one of the coefficients we can determine the others. Since these relations are independent of temperature they should also hold, when the system leaves thermal equilibrium. When Einstein tried to explain the Planck spectrum without stimulated emission, he only got the Wien law. Since hν k B T one has n 1 /n 2 e hν/k BT 1 and hence level 2 is only sparsely populated. Consequently stimulated emission is highly suppressed absorption becomes dominant. One can now express the emission and absorption coefficients in terms of the Einstein coefficients. n 2 A 21 is the rate of emitting transitions per time and volume, we multiply with hν to get from number density to energy density and divide by 4π to get per solid angle : j ν = hν 0 4π n 2A 21 φ(ν) and similarly we treat the absorption (which is proportional to J ν ). α ν = hν 0 4π φ(ν)(n 1B 12 n 2 B 21 ) Stimulated emission is here actually treated as negative absorption. BTW, stimulated emission is the basis for the laser pointer and the laser sword... who would have guessed that this was derived from boring black body radiation.