Radiationtransfer Theory

3.1 Introduction

The production of radiation by bodies around us is usually from what are essentially blackbody surfaces, or surfaces that absorb most of the thermal radiation incident on them. Planetary atmospheres are usually non-ideal blackbodies, unless they contain high amounts of absorbing molecules. Radiation that is emitted by a surface or an atmospheric layer undergoes absorption and/or scattering, by the molecules that constitute the atmosphere, as it is transferred through the atmosphere. Over very large distances radiation is also diluted by the enlargement of space surrounding the radiation source, according to the inverse square law. Absorption and emission by a surface take place over the projected area of the surface in the direction of the radiation. Hence, a planet absorbs solar radiation basically as a disc and emits thermal infra-red radiation as a sphere.

The emission from a blackbody is described by the laws of Planck, Wien and Stefan-Boltzmann. The first gives the spectral distribution of the photons, the second the location of the wavelength of maximum emission and the third gives the dependence of the total amount of radiation emitted on temperature. As the temperature increases, photons at progressively higher energies are emitted. In planetary atmospheres, natural emission is thus restricted to the infra-red whilst absorption can occur at all wavelengths from gamma rays to radiowaves. For thermal infra-red radiation we need to include absorption and emission in the equation of radiation transfer. For the incoming solar radiation we include only absorption and scattering by atmospheric molecules and particles (clouds, aerosols).

We now present some simple classical methods for solving the radiation-transfer equation, as these methods form the basis for obtaining and understanding the radiation field within a planetary atmosphere.

3.2 Basic definitions

3.2.1 The inverse square law

Let us assume that at the centre of a virtual sphere of surface area A there is an infinitesimally small body that is radiating isotropically in all directions. Then, the fraction of the total emitted radiation that is transferred through an element

FIG. 3.1. Radiation sphere with radius R, zenith angle 6, azimuthal angle 0, minor-circle radius r, and element of solid angle dû.

of area dA on the sphere is dA/A. The emitted radiation can be regarded as consisting of small conical bundles of rays with the base of the cone having an element of area dA. The cone of radiation comprises a solid angle, d^, measured in steradians (sr), and given by dl = sin dddd^,

where 0 is the zenith angle and $ the azimuth angle, as defined in Fig. 3.1. In polar co-ordinates, the polar axis defines the zenith angle, whilst the equatorial axis defines the azimuth angle. The radius of the sphere is R, whilst that of a minor circle on the sphere is denoted by r. The element of area dA is then given by dA = (rd$)(Rd0) (3.2)

and hence dA = R2 sin 0d0d$ so that the total area of the sphere is c 2n c n

Thus the fraction of the total radiation transferred through dA is given by dA 1 d^ , s

Thus, the surface of the radiation sphere can be divided into units of steradians with a total 4n (sr) for the whole sphere.

FlG. 3.2. The inverse square law, based on an emitting central sphere of radius Ri and a receiving sphere of radius R2. Surface-area elements for each sphere are also shown.

At the centre of a virtual sphere of radius R2 let us place a spherical body of radius R1 that is emitting isotropically in all directions, as shown in Fig. 3.2. Then the emitted radiation from an element of area AA1 of the central sphere is incident on an element of surface area AA2 on the external virtual sphere, where AA2 > AAi. If the number of emitted photons per unit area is conserved within the solid angle subtended by AA1 and AA2, then the radiation per unit area at a distance R2 from the centre of the virtual sphere is diluted according to the ratio

3.2.2 Radiance

The monochromatic radiance, I\ is defined as the energy (J) emitted by or transferred through an element of area 5A (m2) in a direction defined by the unit vector n within an element of solid angle d^ (sr) per second (s) within a wavelength interval SA about A. If the wavelength is measured in ¡m then the units of the radiance are

where we include the subscript A on the wavelength to distinguish it from the dimensions of the emitting element of area. If we denote the direction that is perpendicular to the infinitesimal element of area, taken to be planar, by the unit vector k, then the projection of the element of area in the direction of the radiation is SA(k • n). Hence, the projected area is reduced as the radiation direction moves towards the horizon according to cos 0, where 0 is the angle

FiG. 3.3. Projection of the emitting area in a direction that makes an angle 6 to the normal.

between the perpendicular direction and the direction of the radiation, as shown in Fig. 3.3. The total energy (J) that is emitted or transferred through the element of area in the direction n is then

in a time interval St, within the spectral interval SX, within solid angle Sl.

3.2.3 Mean radiance and flux

The monochromatic mean radiance at a point in space r is defined as the radiance averaged over all directions according to

The monochromatic radiation flux (or irradiance) is the radiance emitted by or transferred through an element of area, 5A, located at a point r averaged over all directions, accounting for the projection of the surface area according to fx(?,t) = j> Ix(r,n,t)cosddQ (3.9)

and has units W m~2 ^m^1. We can perform the above integral by using eqn (3.1) to express dil in polar co-ordinates and on setting ¡j, = cos 0 we have

2n 1

We note that if the radiance is isotropic then the integral vanishes for an element of area that is simply transferring radiation, as in the case within an atmosphere, and we have no net flux crossing the area. If we have a surface emitting isotrop-ically to the upper hemisphere only, then the integral reduces to h(?,t) = nI\(r, t), (3.11)

where the n has units of steradians. We may integrate the monochromatic flux over all wavelengths to obtain the total flux (or total irradiance, W m~2) emitted by a surface f (r,t)= n Ix(r,t)d\. (3.12)

Within an atmosphere the total flux incident on an element of volume is obtained by integrating over the surrounding sphere f (r, t) = 4n Ix(r,t)d\. (3.13)

3.2.4 Luminosity

The monochromatic luminosity Lx, of an element of surface area emitting or transferring radiation, is the monochromatic flux times the surface area according to

For an element of area that emits or transfers radiation homogeneously

and the total luminosity, L, for isotropic radiation is

In the case of an emitting spherical surface of radius R, the total surface luminosity is

3.3 Blackbody radiation

A blackbody is defined to be a perfect emitter and absorber. For a perfect black-body solid surface or a totally opaque atmospheric layer, the radiance is isotropic and emission is homogeneous over the entire surface area, characterized by a homogeneous surface temperature T(K). For a planet or star with an absorbing-emitting-scattering atmosphere the radiation emitted to space may not be ideal blackbody radiation.

3.3.1 Planck's law

The radiance of an ideal blackbody is given by the Planck function, in terms of wavelength A as

where c is the speed of light in a vacuum, h is Planck's constant and h is Boltz-mann's constant. In terms of frequency (v = c/A) it takes the form

We note that the two forms of the Planck function are related by

which expresses the fact that the emitted energy per sr per unit area per s is the same over the same energy interval.

3.3.2 Stefan-Boltzmann law

The total flux (W m-2) emitted by a blackbody is given by eqn (3.12) by replacing the radiance by the Planck function f (T )= W BX(T )dA. (3.21)

On setting hc/AkT = y and integrating we have where the integral is equal to n4/15, which gives the simple result f (T) = aT4 (3.23)

known as the Stefan-Boltzmann Law, where a is the Stefan-Boltzmann constant

2n5k4

which is equal to 5.67x10~8 J m~2 K-4 s-1. For a star or planet we can assign an effective blackbody temperature, Teff, so that the total luminosity of the body is given by

where R is its radius.

0 5 10 15 20 25 30 35 40 45 50 Wavelength (|jm)

FlG. 3.4. The variation of the Planck function with wavelength and temperature.

3.3.3 Radiation energy density

From the conservation of continuity, the radiation flux can be expressed as flux (W m~2) = density (J/m3) x velocity (m/s), (3.26)

within a blackbody the flux of photons into an element of volume from the surrounding sphere is 4 aT4 hence the radiation density, e, is

4aT 4

3.3.4 Wien's displacement law

The variation of the Planck function with wavelength and temperature is shown in Fig. 3.4. The wavelength, A*, in ¡m, corresponding to the maximum emission of radiation, is inversely proportional to the blackbody temperature according to Wien's Law

2898

The wavelength A* is given by the condition that dB\(T)/dA = 0. As the black-body temperature increases, photons at higher energies become more numerous, since the photon energy hv = hc/A. The peak in the Planck function moves to higher wavelengths as the temperature decreases, and we note from Fig. 3.4 that the Planck function at the lower temperature is always below that at the higher temperature. We also note the rapid rise in the peak value of the Planck function e c

Table 3.1 Radiation type, wavelength range and corresponding order of magnitude blackbody temperature according to Wien's Law.

Radiation

Aa - A6 (/ail)

T( K)

Energy-level transitions

Y-rays

1(T6 - 1(T4

10«

nuclear

X-rays

10-4 _ IQ-2

106

internal electron

ultraviolet

10-2 _ 0.39

104

external electron

visible

0.39 _ 0.75

103

external electron

near-infra-red

0.75 _ 5

103

molecular vibration

infra-red

5 _ 40

102

molecular vibration

far-infra-red

40 _ 400

10

molecular rotation

microwave

400 _ 104

10-1

molecular rotation

radiowave

104 - 107

1(T4

spin reversal

with temperature. Table 3.1 gives the type of radiation and the corresponding typical blackbody temperature for its production.

3.4 Absorption, emission and scattering

Atmospheric molecules can absorb, emit and scatter radiation. Particles such as cloud droplets or ice crystals and aerosols (dust of various chemical composition including water) also play an important role in radiation transfer. During scattering, a photon can change both its direction and wavelength. For planetary atmospheres, absorption and re-emission is coherent in wavelength, as is scattering by particles. Scattering plays a major role in the transfer of solar radiation through a planetary atmosphere by changing the photon direction. An atmosphere may have a physical depth but for photons it is the physical depth divided by the mean free path before absorption and scattering that determines their transfer through an atmosphere.

3.4.1 Absorption and emission

We define absorption as the process by which a particle (electron, atom or molecule) interacts with a photon resulting in the photon energy being transferred to the thermal energy of the atmosphere containing the particle. The inverse process is emission. This process allows energy from one part of the atmosphere to be deposited in another part, possibly far in optical terms. This transfer of energy between layers of an atmosphere has the implication that the system is not closed and so at best only local thermodynamic equilibrium (LTE) can prevail. Strict thermodynamic equilibrium (TE) requires the system to be closed to the outside environment. We shall discuss concepts of TE, LTE and non-LTE in a later section.

Examples of absorption and emission processes include:

Photoionization, or bound-free electronic transitions, whereby a photon is absorbed by an atom resulting in a free electron. The photon energy is taken up by the energy required to free it from its bound state within the atom with the remaining energy going to the kinetic energy of the electron. Photoionization can also occur in stages, whereby an electron is first excited to a higher bound state followed by collisional excitation beyond the ionization energy limit. A free electron can undergo the inverse process to photoionization, radiative recombination, by the emission of a photon and its capture in an atomic bound state. This emission process results in atmospheric thermal energy being converted to radiative energy.

Bound-bound electronic transitions followed by collisional de-excitation. The electron is excited to a higher energy level, the highest being the ionization limit, followed by collision of the atom with another particle resulting in the transfer of the initial photon energy to the thermal energy of the atmosphere. The reverse process is radiative de-excitation, whereby an electron falls to a lower-energy bound state via the emission of a photon.

Free-free electronic transitions whereby a free electron absorbs a photon and changes its trajectory relative to an ion so as to move further from the ion. This is in fact another example of a change in an electric dipole required for photon-particle interaction. The result is that the electron is accelerated because it has gained energy. This is a result of quantum mechanics, as classically an accelerating charge emits photons. The reverse process is free-free emission or Bremsstrahlung, whereby an electron is decelerated along its trajectory relative to an ion with the loss of energy appearing as a photon.

3.4.2 Scattering

We define scattering as the process by which a photon, through its interaction with another particle, can alter either its energy or direction, or both. If the photon energy is not altered significantly then we refer to the photon-particle interaction as being elastic with insignificant transfer of energy to the thermal energy of the atmosphere.

Examples of elastic scattering processes include:

Redistribution scattering, whereby a bound electron is photo-excited to a higher bound state followed by radiative de-excitation back to its original bound state. The emitted or outgoing photon may have undergone a change in energy and direction relative to the absorbed or incoming photon, but the energy change is insignificant. This is essentially elastic scattering, as the energy difference arises from the spread or broadening of the electronic bound states that are not discrete energy states. If they were, then the redistribution scattering would be coherent, i.e. the incoming photon energy and the outgoing photon energy would be identical. However, there is normally a small spread of allowed energies of an electronic bound state due to such broadening mechanisms as natural, Doppler or collisional, which will be discussed in a later section. An extreme example of redistribution scattering is fluorescence, whereby a photon is absorbed by an atom resulting in an electron excited to a higher bound state followed by radiative de-excitation in stages down to some lower-energy level with the emission of several photons of lower energy to that of the incoming photon.

Rayleigh scattering, whereby a photon interacts with a relatively small particle, such as an atom or molecule, as though the particle is a solid body that scatters the photon without the particle undergoing recoil. The scattering process is coherent in energy but the photon direction is modified. The scattering becomes more effective towards smaller wavelengths.

Mie scattering, whereby a relatively large particle, such as a water droplet or dust particle, scatters the photon in a different direction to its incoming direction without a change in photon energy.

Thomson scattering, whereby a photon interacts with an electron without the electron undergoing recoil. This holds for photons with low energies, usually below that of X-rays.

Examples of inelastic scattering processes include:

Compton scattering, whereby a high-energy (X-rays and above) photon interacts with an electron that undergoes recoil. The photon-scattering process is thus incoherent and the outgoing photon energy is significantly different (lower) from that of the incoming photon.

Inverse Compton scattering, whereby an electron of extremely high energy (rel-ativistic) interacts with a photon and transfers part of its energy to the photon. The outgoing photon has thus increased its energy significantly, for example conversion of an ultraviolet photon to an X-ray photon.

3.4.3 The extinction coefficient and optical depth

Let us consider an element of volume V (Fig. 3.5) containing particles each having an effective cross-sectional area, s\, for absorption and scattering of photons of wavelength A. We then define the monochromatic extinction coefficient, \\ as the total effective cross-sectional area, ns\, for absorption and scattering of photons of wavelength A by all particles, n, per volume V at some location r at time t along the direction of the incoming photons. If the photons traverse a distance dl, then the effective volume for extinction, or the reduction in the volume within which photons of wavelength A can be transmitted, is dV\ given by

FIG. 3.5. Volume extinction by an element of volume. The fraction AI\ of the radiance represents those photons that have been either scattered out of the incoming solid angle, dû, about the original direction, n, or absorbed by the particles.

Thus, the radiance, IX, which is a measure of photons per unit area, is reduced by dIX after a distance dl within the volume V according to dI\ dVx

IX V

If the photons travel a distance L then the above equation can be integrated to give Lambert's exponential decay law for photons ix = IoX tx, (3.31)

where IoX is the incoming radiance, tX is the transmission coefficient given by tX = e~TA (3.32)

and tx is the optical depth defined by rx = f xx(l)dl, (3.33)

in differential form.

Thus, if photons of wavelength A traverse a distance L then the probability of not being removed from the initial direction is given by tx. Thus, 1/xx represents a characteristic length, or length constant for the removal of photons of wavelength A along their path, and is thus a measure of the mean free path of photons traversing this element of volume. Thus, the optical depth is the physical distance traversed by the photons in units of mean free path. We thus define the location of the effective optical surface of an atmosphere at an optical depth near unity (e.g. the solar photosphere).

For atmospheres without significant macroscopic motions to modify the random thermal motions of the absorbing/scattering particles, x is isotropic. It has two components, an absorption component, k and a scattering component, a x\(r,t) = Kx(r,t)+ax (r, t). (3.35)

Hence, the probability of a photon undergoing a scattering event as opposed to an absorption event is called the single-scattering albedo, w, given by ux(?,t) = ax (r,t)/xx(?,t). (3.36)

3.4.4 Volume emission and Kirchoff's law

The volume emission, r/X is defined to be the energy emitted at wavelength A, in direction n, per unit volume per steradian per second per unit of wavelength, so that the energy emitted by an element of volume is dE = nX (r,n,t)dAd/d^dAdt. (3.37)

For a blackbody, this can be written also in terms of the emission coefficient, e, representing the effective emission surface per unit volume dE = ex(r, t)BX(r, n, t)dAd/d^dAdt. (3.38)

In the absence of scattering, the amount of energy absorbed by an element of volume can be written equivalently dE = KX(r, t)IX(r, n, t)dAd/d^dAdt. (3.39)

If the element of volume is within a closed system in thermodynamic equilibrium (TE), then IX = BX, and the radiance is given by the Planck function, the radiance of a blackbody. In a state of TE all processes are in equilibrium including radiative equilibrium (RE), and so the emitted and absorbed energy by the element of volume are equal nX(r,n,t) = KX(r,t)lX (r,n,t). (3.40)

Thus, for a blackbody, the emission coefficient is identical to the absorption coefficient eX(r,t) = KX(r,t). (3.41)

This is known as Kirchoff's Law. In the presence of blackbody volume emission, the total volume emission can be written as the sum of a thermal and a scattering component r/X (r,n,t) = KX(r,t)BX (r,t) + nXscat (r,n,t). (3.42)

The thermal component is isotropic, whilst the scattering component in general is non-isotropic.

3.4.5 The source function and redistribution

The ratio of the total volume emission to the total volume extinction is called the source function

In the absence of scattering, we only have thermal emission and the source function is equal to the Planck function. Thus, for thermal radiation Sx = Bx. More generally, we can write the source function as the sum of two components in terms of the single-scattering albedo

Sx(r,n,t) = (1 - wx(r,t))Bx(r,t) + wx(r,t)Sx8cat(r,n,t). (3.44)

The scattering source function is generally non-isotropic and depends on the incoming radiance and the scattering mechanism or photon-redistribution process. Photon redistribution in energy and direction can occur as a result of a bound-bound absorption-emission process or scattering of a photon by a particle. This process is described by the redistribution function, R, which gives the joint probability density for redistribution of photon energy and direction. It is thus meaningful to discuss photon redistribution in terms of photon frequency, v, rather than photon wavelength. Thus, the joint probability of an incoming photon in direction n and frequency v being absorbed and re-emitted or scattered in direction n with frequency v is m > /.\ j dn' dn

R(v ,n \v, n)dv dv--, v y 4n 4n with the normalization condition

COO POO

This strictly holds if the scattering particle does not undergo inelastic collisions with the other particles of the atmosphere, between the time of absorption and emission.

The scattering-volume emission in terms of the redistribution function can be written as

3.4.6 Limiting forms of the redistribution function

Generally, the redistribution function is very complex but there are two limiting simple forms used, based on its physical application. For atmospheric molecules (Rayleigh scattering), aerosols and cloud droplets (Mie scattering), frequency redistribution is not important and only photon redirection is considered. For spectral line absorption-emission, arising from bound-bound transitions, the reverse is true. Particle scattering is thus considered to be coherent in frequency and the scattering coefficient, jv can be a complex function of frequency, depending on the size and physico-chemical properties of the particle. For bound-bound absorption-emission electronic transitions the scattering process is usually taken to be isotropic with only frequency redistribution considered.

3.4.6.1 Isotropic scattering In a static atmosphere, we may average the redistribution function over all incoming or outgoing photon directions to obtain the direction-averaged frequency redistribution function

4n 1

The direction averaging needs only to be performed either over the incoming photon directions or the outgoing photon directions, as the redistribution function depends only on the angle ê, between incoming and outgoing photon directions given by n • n = cos ê, (3.49)

and we have the normalization condition foo

For an atmosphere with significant macroscopic motions care must be taken to obtain a direction-averaged redistribution function, as Doppler effects become important and one needs to average in the reference frame of the particles and not that of the observer. This is because in the observer's frame, the Doppler effects are dependent strongly on direction, resulting in highly anisotropic scattering.

For isotropic scattering, the redistribution function is independent of ê and so it can be written also as R(v , v) and the scattering component of the volume emission can be expressed in terms of the mean radiance Jv>, defined by eqn (3.8), as p O

nvscat(r,t) = av (r,t) R(v ,v)Jv> (r,t)dv . (3.51)

We may average the redistribution function, R(v ,v), over all outgoing photon frequencies, v, to obtain the absorption profile or probability density for absorption of a photon with frequency v , ^>(v ) from y(v ) = / R(v',v)dv, (3.52)

with the normalization condition

The emission profile, ^(v), or probability density for emission of a photon with frequency v, is defined by

This expression holds if the scattering is elastic (no change in the velocity of the scattering particle) and there is no coupling to other bound states, that is we consider transitions only between two electronic energy levels.

Thus, generally the absorption and emission profiles are not the same. There are special cases where they are the same. From the above equation we see that if the mean radiance is a very weakly varying function of frequency and the redistribution function is symmetric R(v ,v) = R(v,v ), then ^(v) = y(v). Another case arises when collisions between the absorbing particles and all other particles are very frequent (high pressures) so that we have complete redistribution in frequency. This situation will be further discussed later when we consider spectral-line broadening mechanisms.

3.4.6.2 Coherent scattering When frequency redistribution is not significant, we can separate the redistribution function into a probability density for photon redirection, described by a phase function, p, and a probability density for coherent scattering in frequency. For bound-bound electronic transitions the redistribution function has the form

where 5(v — v ) is the Dirac delta function, while for particle scattering, such as Rayleigh and Mie, it has the form

and the phase function is a function of cos ê and is normalized according to j) p(cosê)dQ = 1. (3.57)

For isotropic scattering p(cos ê) = 1 and we can define an asymmetry factor, g, which quantifies the extend of the non-isotropy of the scattering process by i r1

The asymmetry factor has the following values according to the scattering process g = 1 forward scattering

= 0 isotropic or symmetric scattering

= -1 backward scattering.

If, for convenience, we set j = cos then for isotropic scattering p(j) = 1, or for symmetric scattering (Rayleigh) p{jj) = f (1 + p2), the asymmetry factor is zero. For Mie scattering or scattering by large particles, defined by the parameter x = 2nr/A > 1, where r is the radius of the particle, the asymmetry factor has typical values near 0.8. For water droplets such scattering represents essentially forward scattering due to refraction. For such scattering the phase function is complex, however, there are approximate expressions such as the Henyey-Greenstein phase function

The above phase function gives a good approximation for the forward-scattering peak, so that an approximation to the Mie phase function, which for water droplets has a strong forward peak due to refraction and a weak backward peak due to internal reflection, is given by p(j) = &phg(j; gi) + (1 - b)pHo(j; 92). (3.60)

For example, the above equation gives realistic values for scattering by marine aerosols at A = 0.7jm, for the values g1 = 0.824, g2 = -0.55 and b = 0.9724.

3.4.7 Forms of the source function

3.4.7.1 Isotropic and coherent scattering For bound-bound electronic transitions with isotropic directional and coherent frequency scattering, the volumeemission scattering component is given by nvscat = Vv 4>(v )5(v - V )JV, dv , (3.61)

and the total volume emission has the form nv = Kv Bv + Vv w(v )Jv, (3.62)

while the source function is written as

where uv is the single-scattering albedo. For particle scattering, the total volume emission has the form nv = Kv Bv + Vv Jv, (3.64)

and the source function is

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