The opacity kv represents the total extinction cross section per mass of interstellar material. If we now wish to explore the contribution of each individual grain, a change of notation is in order. Since the gas itself contributes only a minor part of the extinction in the interstellar medium, we may write pKv = nd ad Qv . (2.38)

Here nd is the number of dust grains per unit volume.The quantity ad, with the dimensions of area, is the geometrical cross section of a typical grain, taken for simplicity to be spherical. That is, ad = na?d, where ad is the grain radius. The ratio of the actual extinction cross section to this projected cross section is Qv, the extinction efficiency factor. As usual, Qv can be written as the sum of absorption and scattering components, which we will denote Qvabs and QVjSca, respectively.

The empirical form of Qv (or its equivalent Q\) can be obtained from the interstellar extinction curve, Figure 2.7, along with some theoretical input to set the scale. Equations (2.26) and (2.38) make it clear that the wavelength variation of the efficiency factor is just

qxo ax0/eb-v where A0 is any reference wavelength. In the limit of zero wavelength, classical electromagnetic theory gives the result that both Qvabs and Qvsca approach unity, yielding a total efficiency factor of 2. Figure 2.7 shows that A\/EB-V = 14 at the last available data point of A = 1000 A. Theory indicates that the extinction rises only slightly at shorter wavelengths, so we may apply this number as our asymptotic value to conclude that

eb-v

Any theoretical model of grain composition must reproduce this basic connection to observations. For example, the behavior of the extinction curve tells us that Qx varies roughly as A-1 in the optical regime. At longer wavelengths, there is a local maximum near 10 pm, not evident in the figure. There is also a more prominent peak in the ultraviolet, at 2200 A. Finally, grain models must account for the scattering and polarization of the interstellar medium observed at visible and near-infrared wavelengths.

In the context of star formation, there has been much effort to determine Qx at far-infrared and millimeter wavelengths. The general interstellar medium is transparent in this regime, so one turns instead to the observed emission from heated dust clouds. Suppose the total optical depth Ar\ through a cloud subtending an angle AQ is less than unity. Suppose further that the cloud is sufficiently transparent in the optical that AV can be determined through counts of background stars, a technique we will discuss in Chapter 6. Then Equations (2.20) and (2.30) imply that the specific intensity leaving the cloud is approximately

Here, we have ignored the absorption term in (2.20). Application of equation (2.17) shows that the received flux can be written as

where we have also assumed that AQ is small. If the dust temperature is known through other observations or theoretical considerations, equation (2.41) yields Atx. Knowledge of AV, together with Equations (2.16), (2.26), and (2.40) then gives Qx.

The determination of Td and AV is usually problematic, so that the wavelength dependence of Qx in this regime is also poorly determined. It is conventional to write this dependence, for 30 pm < A < 1 mm, as . Here, 3 is a positive number, which is thought to decline from about 1 to 2 over this wavelength range. There is also evidence that 3 is generally smaller in the densest clouds and circumstellar disks, but closer to 2 in more diffuse environments. This difference may reflect physical agglomeration of the grains in the denser regions.6 The cumulative effect of these considerations is that Qx, and hence the opacity, is presently uncertain by an order of magnitude at A =1 mm. We will see later that this circumstance hinders the attempt to measure disk masses.

Returning to the issue of grain structure, most extinction observations can be accommodated if the particles consist of refractory cores surrounded by icy mantles. The cores are rich in silicates, such as the mineral olivine found in terrestrial rocks. Silicates can account for the 10 pm feature through the vibration of the SiO bond. Traditionally, the 2200 A spike has been attributed to electronic excitation of graphite, which has therefore been added as another core material. However, the failure to detect graphite in comets and meteorites has cast some doubt on this interpretation. The mantles consist of a mixture of water ice and other molecules presumably

6 As we have just seen, the opacity of a grain does not depend on the wavelength A of absorbed radiation once the geometric size becomes larger than A. From equation (2.38), the wavelength-dependence of the efficiency factor also vanishes in this limit. Hence, centimeter-size grains have a small exponent ft in the millimeter.

adsorbed from the surrounding gas. Such mantles can persist within cold interstellar clouds, but sublimate once the grain temperature exceeds about 100 K.

How large are the grains? In most models, a radius of ad ~ 0.1 pm is adopted, and this frequently serves as a rough estimate. However, it is clear that a continuous distribution of sizes is necessary to match the extinction data. The most commonly used distribution is that of Mathis, Rumpl, and Nordsiek. Here, the relative number of grains per interval in radius varies as a-3'5, with upper and lower cutoffs at 0.25 pm and 0.005 pm, respectively. Equations such as (2.38) are therefore more correctly written as integrals over the size distribution, but we will not need this refinement. 7

Within our simplified picture of uniform spherical grains, we imagine an HI cloud with hydrogen number density nH. We will later find it useful to know £d, the total geometric cross section of grains per hydrogen atom:

If our hypothetical cloud has a length L along the line of sight, then £d can also be written as a ratio of column densities:

Nd ad Nh where Nd = ndL and NH = nH L. We can evaluate the numerator in this relation by first noting that equation (2.38) can be rewritten, after multiplication by L, as

The quantity Ata/Qa can be found using equations (2.26) and (2.40). Applying the result to equation (2.42), we find f Eb-v\ 2

The ratio of color excess to hydrogen column density in equation (2.45) has been established empirically, through an important set of observations. To measure NH in any region, we take advantage of the fact that O and B stars located behind clouds can excite electronic transitions in the intervening hydrogen. For diffuse clouds, the stars are still visible but have additional absorption lines superposed. These lines are in the ultraviolet and can only be observed above the Earth's atmosphere. In 1972, an ultraviolet spectrometer aboard the OAO-2 satellite first measured the Lya transition in the spectra of 69 O and B stars. The depth of the absorption line translates into a hydrogen column density. In addition, the B and V magnitudes of these same stars were observed in order to ascertain their color excess. The two quantities are well correlated, with a linear relationship:

7 The concept of an average grain size is useful because a number of important effects increase with the particle radius. Such effects include the extinction of starlight and the catalysis of H2 formation (Chapter 5). In these cases, the properly weighted average radius is not far below the upper cutoff of the distribution. An exception is photoelectric heating (Chapter 7), which is so efficient at small radii that a full integration is necessary.

Applying this result to equation (2.45) finally yields

This evaluation of £d allows us to obtain a convenient expression for in terms of the standard extinction curve. Referring to equation (2.38), we first write p as mHnH/X. We then solve this equation for the opacity to find

mH nH

Using the definition of £d and its numerical value, we have ka = 420 cm2 g-1 Qx

= 59 cm g —- , eb-v where we have also employed equation (2.40).

It is also instructive to estimate fd, the mass fraction of the interstellar medium contained in grains. Within the picture of uniform spheres, this fraction is fd = ^ M , (2.49)

where pd, the internal grain density, is about 3 g cm-3. The number fraction nd/nH is just Tjd/na?d, which is 3 x 10-12 for ad = 0.1 pm. We thus find for the mass fraction d ~ 3/i,mH (2.50)

Since this number matches the metallicity of the gas, we confirm that a large proportion of heavy elements must be locked up in solid form.

Another important aspect of grains is their ability to polarize radiation. Consider the visible reflection nebulae surrounding bright, young stars. Here, stellar photons travel unimpeded through gaps in patchy cloud gas until they encounter a grain and are scattered in our direction. Prior to the scattering event, the incident electric field vector E oscillates randomly within the plane normal to the propagation direction n (see Figure 2.12). Now focus on radiation that scatters into directions 90° from n, such as s or s' in the figure. These new directional vectors define their own normal planes. The scattered field E only oscillates along the line that is the projection of the new plane and the old. Hence, this radiation is linearly polarized. Scattering into other directions, such as s'', results in partial polarization. That is, E again oscillates along two orthogonal lines, but with unequal amplitude.

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