R Figure 2.10 Attenuation of starlight by the interstellar medium. The point P, located a distance r from a star of radius Rt, receives a specific intensity I\ (r). This intensity is reduced from the value I\(Rt) emitted isotropically from the stellar surface.

Figure 2.10 Attenuation of starlight by the interstellar medium. The point P, located a distance r from a star of radius Rt, receives a specific intensity I\ (r). This intensity is reduced from the value I\(Rt) emitted isotropically from the stellar surface.

Our next step is to use equation (2.17) to find the specific flux received at P. In the present case, n « 1, and the small solid angle AQ subtended by the star is nR^/r2. Thus

Imagine now that the same star were located at some other distance ra from P, and that there were no intervening extinction. Writing the received flux in this case as Fx (ra), we have simply

If we now divide equation (2.22) by equation (2.23), the result can be written in the form

-2.5 log(r) = -2.5 logFa*(r0) + 5 log(^) + 2.5 (loge) Ata . (2.24)

Referring to Equations (1.2) and (1.3), we let ra equal 10 pc. We then add the constant mx0 to both sides of equation (2.24) to find mx = Mx + 5 log () + 2.5 (loge) Ata • (2.25)

V10pcy

Comparison of this equation with (2.12) yields the desired result:

Thus, the two measures of extinction actually give very similar numerical values. 2.3.4 Blackbody Radiation

Let us now examine quantitatively the grains' thermal emission. Imagine surrounding a portion of interstellar gas, with its admixture of grains, by a container whose walls are maintained at temperature T. Suppose further that these walls absorb all the photons impinging on them, while the gas is transparent to this radiation. Then the heated walls will generate their own photons, and the interior of the container will be filled with radiation that is in thermal equilibrium with the walls. This means that the distribution of photons among the available quantum states is the most probable one consistent with the free exchange of energy between radiation and matter. The radiation energy density under these conditions is given by the Planck formula:

8nhv3/cC3

The so-called blackbody radiation we have just described is also isotropic, i. e., the specific intensity Iv is independent of direction. From equation (2.18), we have Iv = cuv/4n. The specific intensity in this case is given the special symbol Bv and is a function of temperature alone:

to obtain

Within our hypothetical container, the spatial uniformity of the radiation field implies, from equation (2.20), that the emissivity of the matter obeys

where we have indicated that only the absorption component of kv counts here. Now remove the container walls. The same matter must emit thermally at precisely the same rate. In applying equation (2.30), we must take care to use Td, the temperature of the dust grains, which may be quite different from the temperature of the surrounding gas (see Chapter 7).5

Figure 2.11 plots the important function Bv (T) for several values of T. We see that increasing T raises the intensity at all frequencies but maintains the shape of the curve. By making the substitution x = hv/kBT in equation (2.28), the reader may verify that Bv (T) reaches its maximum at x0 = 2.82, so that vmax _ xo kB

Similarly, from substitution of y = hc/XkBT into equation (2.29), it follows that B\(T) peaks at the wavelength Amax, where

5 Equation (2.30) is a statement of Kirkhoff's law; see also Appendix E. Note that we adopt the convention that a subscript d denotes dust, while g denotes gas.