Determining either rCR(H2) or rCR(HI) still requires measurement of the respective ionization rates, Z(H2) or Z(HI). Here, the main impediment is our ignorance of the interstellar flux of low-energy cosmic rays. We have noted, however, that the ionizations are the first steps in a network of ion-molecule reactions that lead to the formation of more complex species. We may therefore use the observed abundances of selected molecules to infer the ionization rates indirectly.
One such molecule is OH, which, along with hydrogen itself, can sometimes be detected in diffuse clouds through its ultraviolet absorption lines. In this manner, observers have measured the typical number density of OH relative to HI as 2x 10-7. (Compare Table 5.1.) The sequence of reactions forming OH from atomic oxygen and hydrogen begins when the H+ created by cosmic rays encounters an oxygen atom from the ambient gas:
Note that the H+ may also recombine with free electrons before such charge exchange can occur. We will simplify the analysis by ignoring this possibility. As illustrated in Figure 7.3, the production of O+ initiates a chain of reactions with H2:
The H3O+ formed in the last reaction cannot gain any more hydrogen atoms, and instead undergoes dissociative recombination with two possible outcomes:
The first reaction in (7.10) is known from laboratory measurements to occur with a relative probability p1 = 0.75.
Since each ionization of a hydrogen atom by a cosmic-ray proton leads to an OH with probability p1, the volumetric production rate of OH is p1 Z(HI) nHi. In steady state, this rate is balanced by the destruction of OH. The molecule can react with ambient ions such as C+, but more frequently dissociates from ultraviolet radiation penetrating the cloud interior. Writing the characteristic photodissociation time as rphoto, we balance creation and destruction to find the desired expression for the rate of cosmic-ray ionization:
In the numerical evaluation, we have inserted the theoretical result Tphoto = 2 x 1010 s, relevant for the diffuse clouds of interest. Note that our empirical value of Z(HI) includes both direct ionizations by protons and secondary ionizations by ejected electrons. Substituting this value into equation (7.7), we find r0R(ffl) = 1 x 1(T13 (^^=3) events-1 (7.12)
It remains to determine the corresponding rates for molecular hydrogen. Theoretical calculations show that the probability of a hydrogen molecule being ionized by a cosmic-ray proton is 1.6 times the atomic value. Utilizing the enhancement factors from secondary ionizations, we have
so that rCR(H2) = 2 x 1(T13 (jô^zâ) eVcm^s-1 . (7.14)
Comparison with equation (7.12) gives the simple result that, whether the gas is in atomic or molecular form, the cosmic ray heating rate is the same as measured per hydrogen atom.
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