Description of radiation field

Description of radiation field Qualitatively, we know that characterization should involve energy/time frequency all functions of x,t. direction We also now that radiation is not altered by passing through empty space. Flux of energy varies in empty space and therefore cannot be the fundamental quantity. L energy time Ex.: Fs = L 4r ˆr r 2 Energy per solid angle works. Assume solar surface emits uniformly Each unit of area on Sun emits Fixed solid angle receives emission from area of r 2. For reference: A = 1 r 2 energy flux solid angle, time ( r r 2 A r )+ independent of r 1 ( r cos A cos )+ 1 r cos A

Intensity or radiance lim I ( x,t, ˆn )= 's 0 where t = time = wavelength A = area ˆn E t A = solid angle With I defined this way, radiative transfer in empty space reduces to di = 0, s = distance in ˆn direction (along beam) ds Energy flux In direction ˆn, monochromatic flux is F ˆn = I ( nˆ )ˆ n ˆn d 4 Total flux Fˆn = 0 F ˆn d Comments: *Notice that if I ( nˆ )= I ( nˆ ) then = 0 Fˆn * I ( nˆ )ˆ n is a vector and is a sum of vector components Fˆn F = ê Fê1 1 + ê Fê2 2 + ê Fê3 3 is vector energy flux.

*Isotropic radiation I ( x,t, ˆn ) I ( x,t ) No net flux by symmetry. Flux in one direction across surface is of interest. F + = I ( ˆn )ˆn + ˆn d( ˆn ) 2 2 1 = I cos 0 0 d ( cos)d = I Notice I < 2 I (whole hemisphere). This result is of special interest because thermalemission from a surface or isotropic scattering from a surface give this geometry. + direction Interaction with matter - Absorption A medium can absorb, emit or scatter radiation. Under absorption alone one expects Energy loss along = Beam *absorption strength *n *s beam distance s intensity per particle # particles vol 1 Since ns = particles, need absorption strength in area I = I ˆ a n s Salby notation

Thus di = ˆ I a n I a I Units = length 1 The factor is the absorption coefficient. This attenuation equation is called Lambort's Law. Its integral is di = ds I ln I = s ds + const. I = I ( s 0)e s a ds 0 The "optical path" or "optical depth" u s ()= s a ds 0 has no units. s I () ( ) = s eu I 0 () Transmissivity Comment: Absorption strength is ecpressed in a variety of units. Fundamental combinaion is u. di I = ds = du = ˆ a n ds u is always (strength measure) (gas amount along path) area Strength might be particle, area optical path,. Be careful mass km.amagats

Planck intensity spectrum A radiation field inside a container with walls at uniform temperature comes into thermodynamic equilibrium. (It is assumed that the walls have a continuous spectrum of energy transitions that can happen.) This spectrum is the Planck Function. 2h c 2 energy = 5 exp + hc (area)(time)(wave length)(steradian) 1 k T This is energy per unit wavelength. To evaluate,energy per unit frequency use E = = = d d, = c 2h 3 = c 2 exp h 1 k T In practice the first and second radiation constants are used to numerically evaluate. For example: c = 1 5 exp c 2 T 1 Max of is at T = 0.29 cm K. Max of is at T = 0.51 cm K. c 1 = 3.7419 10 16 W m 2 c 2 = 1.4388 10 2 m K

Brightness temperature Thermal emission spectra are often resented in units of brightness temperature. Observe I ( ) radiance units Set I equal to the Planck Function at unknown T b ( T b,)= I Invert, solve for T b ( ). T b ( ) gives the temperature, at each wavelength, that a black surface would need to have in order to emit the observed intensity. ( )

Stefan-Boltzmann Law Many surfaces emit approximately the same way that a port in a black cavity would emit. I = ( T,) independent of direction F = F = d = T 4 0 = 5.67 10 8 Wm 2 K 4 Example: Estimate the temperature of a planet. Power received from = a 2 1 A 2 4r = a 2 F s ( ra), F s 1368 W m 2 L Power emitted by planet = 4a 2 T 4 T 4 = 1 A 4 F s T = 254 K = 236 W m2 ( ) A 0.31 a

Equation of transfer Thermal emission In the presence of absorption and emission, di ds = I + S Where S is a source term due to emission by the medium. In a region many optical depths across, the exterior boundaries will become unimportant. Then if the region is isothermal, the radiation field will be in thermodynamic equilibrium, with I = B. Then di ds = 0, and I + S = 0 S = B di ds = ( I B ) Emission coefficient = absorption coefficient is Kirchoff s Law.

Emission and scattering Denote e exxtinction coefficient. In general e i n i i = monochromatic cross section (lenght) 2 n i = number density of constituent i. Let o be fraction of extinction that scatters into a new direction and is not absorbed. o single scattering albedo. Governing equation di ds = I e ( + J ), ( scat) J ( ˆn )= o P( ˆn, nˆ ) I ( nˆ ) d 4. 4 By o definition, P( ˆn, n ˆ) d 4 = 1 4 ( emission) What is J? Expect C B,if LTE. di ds = I e + 0 P( ˆn, nˆ ) I ( nˆ ) d 4 + C B Deep inside homogeneous region I B (isotropic) and di = 0. Thus ds -B + 0 B P( ˆn, nˆ ) I nˆ ( ) d 4 + C B = 0 Ex. Forward scattering nˆ ˆn

C = 1 0 Summary: Kirchoff again 1 di e ds = I + 1 ( 0 )B + 0 P( ˆn, nˆ ) I nˆ 4 ( ) d Intensity change Extinction Emission Scattering Formal integral in absence of scattering Let e ds = dx optical thickness s = 0 s x x =0 I di = I dx B ( T ) (no scattering) e x d e x I dx ( )= B x ( ) dx e x I x = e x 0 B 0 X e x I I x X ( = 0)= e x ( B 0 X ) dx I ( X = 0)= I ( X )e x + x e x B X Intensity emerging Attenuated incident intensity 0 ( ) dx Accumulated emission along path

Absorption spectrum Transitions In planetary atmospheres transitions between discrete energy levels are of most importance. In stars free-free or bound-free transitions give continuum absorption. absorption lines dominate. h = E Exception: UV and EUV absorption in upper atmosphere. Transitions are Electronic visible Vibrational IR & combinations Rotational far IR Examples of spectra are shown in Salby and in viewgraphs. Units Spectrscopists usually work in frequency units expressed in wave numbers = c ( ) = c s1 = c = 1 (lenght1,usually cm 1 ) wave number Handy identity: ( microns)= 10,00 cm 1 ( )

Lorentz line profile In the lower atmosphere, collisional broadening is the most important factor producing frequent dispersion of absorption. Phase coherence of emitted photon is destroyed by collisions. t = time between collisions t Spectrum is broadened by 1 t. 1 1 wave number units c t To estimate, l mfp A n = 1 1 t = l mfp = An A ~ ( 4a 0 ) 2 o, a 0 = Bohr radius ~ 0.5 A 10 19 m 2 n ~ 2.69 10 25 m 3 Loschmidt's # ~ 300 m s (sound speed) 1 1 An ~ ~ 3 102 10 19 2.7 10 25 ~3 m 1 t c c 3 10 8 =.03 cm 1 Typical IR frequency = 10 microns = 1000 in 1.03 cm -1 Narrow. Since n, Width pressure. 1000 cm -1

The shape is the Lorentz profile f L = L ( ) 2 + L 2 0 0, f L d = 1 0 The cross section is = S f L ( ), S line strength. The strength is a function of temperature S(T ), and L is usually taken to be L = L0 p T 0 p 0 T n, where p 0,T 0 are STP, and n 1 2 and L0 are tabulated in line atlases. Doppler line broadening ~ 350m s1 ~ c 3 10 8 ~106 1 m s ~10 6 ~10 6 10 3 ~.001 cm 1 Important in upper atmosphere where p small and Lorentz width is small. Doppler shape: D = 2K 0 3 T c m 1 f D = D exp 2 0 0

Voigt line profile Convolve Doppler & Lorentz broadening f ( 0 )= f L ( 0 ) f D ( )d There are efficient algorithms to evaluate it See Grody and Yung. Line atlases See the description of the HITRAN, GEISA atlases. Line strength S(T) involves lower state energy. This is because of state population dependence on T. From L TOU, n T S i ( T ) S i ( T r ) T exp hce i 1 r K B T 1 T r n = 2 linear molecule n = 5 2 nonlinear molecule E i = lower state energy for line i T r = 296 K = reference T.

Plane parallel atmosphere, emission and absorption Transfer eq. in terms of optical depth. d = dz No scattering. di ds = I ( B ) Geometry of ray ds= dz dz cos μ Plane parallel is idealized case = ( z), B = B ( z). Then I = I ( z), 1 di ds = μ di dz = μ di d z \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ds μ di d = I B Solution for upward intensity I ( z,μ)= I ( s,μ)e + B ( )e ( z) s ( ) μ z μ s μ ( z) d μ μ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ z Z = 0

Usually I ( s,μ) B ( T s ). In some cases, I ( s,μ) B ( T s ). Where the emissivity varies spectrally but with wide Bands rather than narrow lines. Mars surface mineralogy Is an example. For now, assume B (T s ). Average over a band In practice a finite is of interest. For heating rates can average over spectrum. For remote sensing the instrument has spectral resolution limitations. Average over small enough so that the Plank function ~ constant. I ( z,μ)= 1 B ( T s )e s μ d + 1 s B ( ) e μ d μ where B smooth B ( T s ) 1 e s s μ d B s = B ( z,0,μ)+ B ( z) d z, z,μ dz ( z, z,μ)= 1 e N.B. Sign changes from d dz μ d ( ) d e d μ μ ( ) 1 dz and from reversal of integration range.