The differential equation that governs the absorption and emission in a layer of gas follows from the geometry of Fig. 17. A uniform cloud ("source") of temperature Ts, depth A, and optical depth ta lies between the observer and a background source at some other temperature T0.

Figure 11.17. Geometry for the radiative transfer equation. The background surface emits with specific intensity I0 and the intervening gas cloud emits thermal radiation with specific intensity Is when it is optically thick. An observer in the cloud at position x, or optical depth t viewing leftward will detect radiation from the cloud atoms at lesser t and from the background source to the extent it is not absorbed by the cloud.

Figure 11.17. Geometry for the radiative transfer equation. The background surface emits with specific intensity I0 and the intervening gas cloud emits thermal radiation with specific intensity Is when it is optically thick. An observer in the cloud at position x, or optical depth t viewing leftward will detect radiation from the cloud atoms at lesser t and from the background source to the extent it is not absorbed by the cloud.

For the immediate discussion, we refer to radiated intensities at some single frequency (in a differential band) without regard to the entire spectrum. In fact, the overall spectral shape may be inconsistent with a single temperature. Hence we discuss intensities without necessarily defining a temperature. Nevertheless, in stellar atmospheres, temperatures can be defined in regions of local thermodynamic equilibrium (LTE). In this case, a higher intensity at a given frequency does represent a higher temperature.

In the absence of the cloud, the observer in Fig. 17 would detect a specific intensity I0 (W m-2 Hz-1 sr-1) from the background source (T0) at the frequency in question. We refer to I0 as the background intensity. If the intervening cloud is in place, it will absorb some of the radiation from the background source. In addition, the cloud will emit its own thermal radiation in the direction of the observer. If the cloud is optically thick, the emerging radiation would exhibit the specific intensity Is characteristic of blackbody radiation at Ts.

Intensity differentials Consider a beam of photons moving in the direction of an observer at some location in the cloud. The differential equation that describes absorption of the photons in a differential path length dx is, from (10.17), dN/N = —an dx, where dN /N is the fractional change in the number of photons in the beam, a (m2) is the cross section per atom, and n (m-3) is the number density of atoms. The fractional change of the photon number will be equal to the fractional change in the specific intensity, giving, d/1

where d/1 is one of two contributions to the total change d/.

The cloud also contributes photons to the beam. The thermal emission originating in the layer at x in dx of the cloud can be described with the volume emissivity j (W m-3 Hz—1). This gives rise to an element of specific intensity from the layer in question which is, from (8.48), j dx d/2 =--(Thermal emission from gas) (11.35)

The sum of these two effects yields the net change in intensity / of the beam, j dx d/ = — /an dx +--(Net change in / in dx at x) (11.36)

This is the differential equation that allows us to find, by integration, the variation of beam intensity as it traverses the material on its way to the observer.

/ntensity variation with optical depth

Rewrite (36) to be a function of optical depth t . Recall the definition of the opacity, k = an/p (10.24), where p (kg/m3) is the mass density. Opacity is the cross section per kilogram of material (m2/kg). Substitute Kp for an into (36) and rearrange,

Kp dx 4nKp

The product Kp or a n is simply the inverse of the mean free path xm with units of (m—1); see Table 10.1. Thus the product Kp x is the number of mean free paths in the distance x for fixed k and p. In other words it is the optical depth t = Kpx, a dimensionless quantity previously defined (10.29). The denominator Kp dx of the left side of (37) is thus equal to dT since k and p do not change appreciably in an incremental distance dx.

The equality (37) demands that the rightmost term have units of specific intensity. Since j is the volume emissivity of our cloud, we define this term to be the cloud intensity, or the source intensity /s, j (W m-3 Hz-1) 4n(sr) Kp (m-1)

This expression has the form of (8.53), the relation between j and / for an optically thin plasma of thickness A, namely / = javA/4n. Here, the mean free path (Kp)—1 plays the role of the cloud thickness A. In our optically thick case, an observer can "see" only a depth of about one mean free path into the cloud. The source intensity (38) is thus the intensity an observer embedded in the cloud would measure if her view were limited by optically thick conditions (t > 1).

It follows from the above considerations that the differential equation (37) may now be written as d I (t )

transfer)

where we express I as a function of t , the optical depth of the cloud in the observer's line of sight (Fig. 17). This is the differential radiative transfer equation (RTE) which may be solved for the unknown quantity I (t), the specific intensity at optical depth t for our chosen frequency.

In (39), I(t = 1) is the specific intensity measured by an observer within the cloud at the depth of one mean free path into the gas. Note that depth is measured from the left edge of the cloud. At t = 0.1 or t = 3, the function I (t ) is the specific intensity at depths of 0.1 and 3 mean free paths respectively. If the entire cloud has optical depth ta (corresponding to thickness A), the function I(ta) is the specific intensity measured by the observer outside the cloud.

The quantity I (t ) is distinct from Is. It includes the radiation from the background source I0 as modified by absorption and emission in the cloud. The background radiation is the "initial condition" we apply to the differential equation (39). The source function reflects the volume emissivity of the cloud itself.

The quantities t, I (t) and Is in (39) are all functions of frequency; namely t(v), I(v), and Is(v). We continue to consider one frequency only and suppress the argument v. The function j, and hence Is, can vary with position in the cloud, i.e., both can be functions of t. This is the case in stellar atmospheres where the temperature varies continuously with altitude. In the following, we consider Is to be a constant throughout the cloud; the important consequences of (39) are well illustrated in this case.

If the gas of the cloud were in complete thermodynamic equilibrium, the radiation and matter would all be in thermal equilibrium at some temperature T ; the specific intensity I(t) would not vary throughout the cloud. In this case, the derivative in (39) equals zero, dI/dT = 0, and the observed intensity I(t) is given by

(Perfect thermal (11.40)

equilibrium)

which is independent of t . Since I (t) is the specific intensity for complete thermodynamic equilibrium, its spectrum must be the Planck (blackbody) function (23). In turn, the source intensity Is must also have a blackbody spectrum.

The solutions we seek are, in general, not for complete thermodynamic equilibrium because they involve a gas at one temperature and incoming photons representative of a slightly different temperature. Also, the limited extent of the cloud implies that radiation is leaving the volume of the cloud, so that complete equilibrium can not exist near the surface. Nevertheless, in solving the RTE one can make the assumption of local thermodynamic equilibrium (LTE).

Under LTE, the matter (e.g., protons and electrons) in a local region is in equilibrium with itself, but not necessarily with the radiation. That is, the matter obeys strictly the Boltzmann-Saha-Maxwell statistics, i.e., (9.14) and (9.15), for the local temperature, but the photon distribution is allowed to deviate slightly from it. Nevertheless, the radiation emitted from the local region follows the frequency dependence of the blackbody function for the local temperature, according to (40). The source function Is for radiation emitted in a local region is therefore the blackbody function (23) for the (matter) temperature of the local region.

One can show that in the solar photosphere, the number density of particles is ~105 times that of the photons. Since every such photon or particle has about the same energy, ~kT, in thermal equilibrium, the energy content is overwhelmingly contained in the particles. They can thus maintain their own temperature and radiate at that temperature in their local region even if photons from a lower and slightly hotter region diffuse up into their region and minimally distort the overall photon spectrum.

Insight into the behavior of I(t) according to the radiative transfer equation (39) can be gained simply from knowledge of the relative magnitudes of I(t) and Is. If I(t) < Is at some depth t, the derivative in (39) is positive which tells us that I(t) increases with optical depth. This is shown as the heavy line in Fig. 18a; note that it lies below the horizontal dashed line for Is. Recall that in our case we hold Is constant throughout the cloud. If, on the other hand, I(t) > Is, then I(t) decreases with depth (heavy line in Fig. 18b). In each case, I(t) moves toward Is and asymptotically approaches it at large optical depth.

At zero optical depth, I (0) is equal to the background intensity I0 because only the background source, and no part of the cloud, is in the observer's line of sight as is clear from Fig. 17. We also see this in both panels of Fig. 18. This obvious result also follows from the formal solution of the RTE to which we now proceed. We will find that the solution naturally provides for absorption and emission lines.

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