Arm

E Ma

Z = A>-= 0.019 ± 0.002 ("Heavy element" (10.42)

A Ma mass fraction)

Equation (41) indicates that ~27% of the total mass (of all elements) is in helium, a value significantly larger than the ratio (0.10) of helium to hydrogen (only) by number. The difference arises because each helium atom has ~4 times the proton mass and because the ratios are relative to hydrogen alone in one case and to the total mass of all elements in the other. When quoting abundances one should take care to specify precisely the quantity being used. The two conventions given here are quite standard, namely numbers relative to hydrogen alone (Table 2) and masses relative to the total mass (40)-(42).

Propagation distances in the interstellar medium

The summed contributions of all the elements in the interstellar medium yield a net cross-section curve for photoelectric absorption that can be used to calculate propagation distances in the Galaxy.

Effective cross section

An effective cross section in the interstellar medium may be calculated from our knowledge of the relative abundances of the elements (Table 2). The result for the UV to x-ray region is plotted in Fig. 7a. (The optical region at ~2 eV is off the plot,

Log photon energy (eV) Log photon energy (eV)

Figure 10.7. Effective photoelectric cross section per hydrogen atom of the interstellar medium versus photon energy, for (a) the entire UV to x-ray range and (b) the x-ray range enlarged. The ordinate is the cross section of a single hydrogen atom enhanced to include its share of the cross sections due to heavier atoms. The cross section at any given photon energy is dominated by the element whose K edge is next below the photon energy. Horizontal dashed lines indicate the cross sections that yield attenuation to e-1 = 0. 37 for Proxima Cen and the galactic center, for a typical particle density in the interstellar medium, nH = 0.5 x 106 m-3. Note that the scales of the ordinates are compressed relative to the abscissas. [<13.6 eV from Cruddace et al, ApJ 187, 497 (1974); 13.6-280 eV from H. Marshall personal communication; >280 eV after Morrison and McCammon, ApJ 270, 119 (1983)]

Log photon energy (eV) Log photon energy (eV)

Figure 10.7. Effective photoelectric cross section per hydrogen atom of the interstellar medium versus photon energy, for (a) the entire UV to x-ray range and (b) the x-ray range enlarged. The ordinate is the cross section of a single hydrogen atom enhanced to include its share of the cross sections due to heavier atoms. The cross section at any given photon energy is dominated by the element whose K edge is next below the photon energy. Horizontal dashed lines indicate the cross sections that yield attenuation to e-1 = 0. 37 for Proxima Cen and the galactic center, for a typical particle density in the interstellar medium, nH = 0.5 x 106 m-3. Note that the scales of the ordinates are compressed relative to the abscissas. [<13.6 eV from Cruddace et al, ApJ 187, 497 (1974); 13.6-280 eV from H. Marshall personal communication; >280 eV after Morrison and McCammon, ApJ 270, 119 (1983)]

immediately to the left.) This plot is a summation of photoelectric curves like those of Fig. 6, for all different elements and taking into account the relative numbers of each species in the interstellar medium, and normalizing to the hydrogen content as described below.

The discontinuities represent ionization energies of the several elements. At the lowest photon energies (in the ultraviolet), the cross section is due to ejection of electrons from the higher levels of the heavier elements, i.e., Al, Si, Mg, S, C. At the K edge of hydrogen, 13.6 eV, the cross section jumps upward by 5 decades! At this energy, the interstellar medium becomes very opaque. The region above this energy is known as the extreme ultraviolet.

Thereafter the cross section decreases approximately according to the -8/3 law (38) except for small jumps due to K edges of the heavy elements. This region is magnified in Fig. 7b with the K-edge photon energies indicated in keV. At cross sections below about 10-26 m2, the interstellar medium of the Galaxy is quite transparent to radiation (horizontal dashed line for galactic center). Thus, as noted earlier, one can do galactic astronomy in the ultraviolet up to 13.6eV(k = 91.2 nm) and in the x ray beyond ~2 keV.

Most of the effective cross section in the x-ray region (beyond the K edge of carbon) is due to the heavy elements indicated. Their smaller numbers are easily offset by their much greater cross sections (per atom) in the regions near their K edges. These same heavy elements in interstellar grains give rise to optical extinction.

The ordinate in Fig. 7, aeff, is the effective cross section per hydrogen atom. Each H atom is assigned its share of the cross section that is due to other elements. For example, we have seen that there is 1 He atom for every ~10 hydrogen atoms in the interstellar medium. The effective cross section per H atom therefore includes the cross section of a single H atom and an additional ~1/10 the cross section of a single He atom. The trace amount of each heavier element is included in the same manner. There is 1 carbon nucleus for each ~2800 hydrogen atoms; thus 1 /2800 of the carbon atom cross section is included, and so on. Thus, aeff = aH + ai fi (Cross section per H atom) (10.43)

i where ai is the cross section of element i and fi is the number abundance relative to hydrogen.

This way of presenting the total cross section, "per H atom", is often quite convenient. Radio astronomers can determine the amount of hydrogen in different parts of the Galaxy by detecting its emission at 1420 MHz and measuring intensities and Doppler shifts. They are thus able to determine densities and locations of hydrogen clouds. The other elements are more difficult to measure and generally are not well known for most regions of the Galaxy. If one assumes that the local (solar system) cosmic abundances are approximately those of the rest of the Galaxy, one can use the cross section aeff of Fig. 7 in conjunction with the radio measurements of Nh to estimate the total absorption for photons from various places in the Galaxy.

Survival distances

The depletion of a photon beam of a given frequency may now be calculated from (20), N(x) = Ni exp(-anx), for a uniform medium of hydrogen number density nH and effective cross section aeff. In this case, a nx ^ aeff nH r = aeffNH where r is the distance to the source of the photons and NH is the column density of hydrogen atoms (m-2) from the observer to the source. Thus, in terms of photon fluxes, the attenuation at frequency v is

p v = exp(-CTeff(v)Nh), (Attenuation in the ISM (10.44)

where oeff(v) is obtained from Fig. 7, and NH may be inferred from the radio data.

Knowledge of oeff and NH yields the ratio of photon fluxes. For an astronomical source, the two fluxes in the ratio may be considered to be those one would measure at the earth, the one in the numerator with the absorbing material in place and the one in the denominator without it, hypothetically. If the flux at the earth is measured, one thus finds the intrinsic (without absorption) flux.

Consider 1.0-keV x rays traveling in the plane of the Galaxy, e.g., from the galactic center to the sun, a distance of ~25 000 LY. A nominal value for the interstellar number density of neutral hydrogen n in the galactic plane is n ~ 0.5 x 106 m-3, but one must keep in mind that the actual value from place to place in the Galaxy can vary by many orders of magnitude, from ~103 to ~1010 m-3. The curve of Fig. 7b, for an energy of 1.0 keV, yields <oeff = 2.4 x 10-26 m2. Thus, the values, nH = 0.5 x 106 H atoms m-3 (Galactic center to sun) r = 25 000 LY = 2.4 x 1020 m (10.45)

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