HEATING OF THE WARM IONIZED MEDIUM BY LOW-ENERGY COSMIC RAYS

At energies $\gtrsim 1\;\mathrm{GeV}$, the cosmic-ray flux in the neighborhood of the Sun is tightly constrained by direct measurement (e.g., Webber 1998; Olive et al. 2014). At lower energies the attenuating effects of the solar wind become increasingly important, and much below $100\;\mathrm{MeV}$ there is little constraint from spacecraft data. Our understanding of cosmic ray acceleration and propagation is also not good enough for us to confidently predict the low-energy fluxes based on theoretical models of those processes. Indirect observational constraints are therefore pivotal.

As cosmic rays propagate through the ISM, they lose energy through various processes—ionization, "pionization" (for hadrons), and bremßtrahlung (for leptons)—being some of the most important (e.g., Padovani et al. 2009; Olive et al. 2014). At low energies, ionization dominates in neutral media and the ionization state of the ISM provides information on the low-energy cosmic ray (LECR) population (e.g., Hartquist et al. 1978; van Dishoek & Black 1986; Federman et al. 1996). Dalgarno (2006) has given an overview of the relevant chemistry and results that have been obtained in dense clouds, diffuse clouds, and the intercloud medium. There are difficulties because ionization balance is influenced by various factors: e.g., the density, column density, and composition of the cloud; its homogeneity; and the ionizing photon background. Nevertheless, in recent years there has been progress from studies of ${{\rm{H}}}_{3}^{+}$ in diffuse molecular clouds (McCall et al. 2003; Dalgarno 2006; Indriolo et al. 2007). The particular appeal of ${{\rm{H}}}_{3}^{+}$ is that the chemistry is simple so the interpretation of the data ought to be fairly robust (but see Shaw et al. 2008). These studies consistently point to large ionization rates—much higher than can be produced by the known GeV cosmic rays—implying high cosmic-ray fluxes at low energies (Indriolo et al. 2007, 2009; Padovani et al. 2009). In this paper we consider the implications of those results for the LECR heating rate of the diffuse, ionized component of interstellar gas, which we refer to as the warm ionized medium (WIM) (Reynolds 2004; Haffner et al. 2009).

The LECR interaction that causes the ionization of neutrals is just Coulomb scattering—electrons receiving a kick from the electric field of a passing cosmic ray—and this process has a larger cross-section when the electrons are free than when they are bound in atoms or molecules. Thus high ionization rates for neutral gas immediately imply large heating rates for ionized gas. We will show that LECR heating alone may suffice to maintain the WIM. This result contrasts with the prevailing view (e.g., Haffner et al. 2009; Wood et al. 2010; Hill et al. 2015) that only UV photons carry enough power to be able to maintain the WIM.

With the data available at present, a wide range of spectral forms is possible for the LECR. Consequently, it is unclear whether one should think of physically distinct low- and high-energy cosmic ray components (e.g., with a different acceleration mechanism) or whether they are different aspects of the same phenomenon. It is also unclear whether the large ionization rates of diffuse molecular gas are caused primarily by cosmic-ray protons or electrons (Padovani et al. 2009), so we consider both possibilities.

Observations of hard X-rays from the inner Galactic disk have also been interpreted in terms of powerful LECR fluxes (e.g., Skibo et al. 1996; Valinia et al. 2000). Although the inferred LECR spectra, and thus the WIM heating rates, are model-dependent, the X-ray evidence is independent of the molecular ion studies and thus provides a valuable check. Furthermore, the X-ray data sample a large volume of the Galactic disk and provide a more representative view of the LECR than local studies.

This paper is structured as follows. In Section 2 we recap the theory of energy losses via Coulomb collisions of individual LECR and we apply the theory to both neutral and ionized hydrogen. In Section 3 we use the measured ionization rates of diffuse molecular gas to estimate the total Coulomb heating of ionized gas in the solar neighborhood, largely independent of the uncertainties in the LECR spectrum. Our estimates are comparable to the heating rate needed to maintain the WIM at a temperature of $\sim {10}^{4}\;{\rm{K}}$. In Section 4 we reinforce that point by referring to published fluxes of LECR in the inner Galactic disk, determined from hard X-ray observations. Those studies imply that LECR heating of the WIM could be much larger than our local estimate; together, these results lead us to conclude that LECR may well be the dominant heat source for the WIM.

The interactions between cosmic rays and molecular hydrogen have been quantified over a broad range of particle energies ($0.1\;\mathrm{eV}\mbox{--}100\;\mathrm{GeV}$) by Padovani et al. (2009). Olive et al. (2014) give a comprehensive review of high-energy processes in different materials. For LECR the dominant energy loss process is Coulomb collisions, treatments of which can be found in many textbooks, e.g., Jackson (1962), Ginzburg & Syrovatskii (1964), and Longair (1981). Our discussion makes use of a simplified calculation in which the expected rate of energy loss of a cosmic ray, of speed $\beta c$, with increasing electron column, N_e, is:

$$\begin{eqnarray}&&-{\left\langle \displaystyle \frac{{dE}}{{{dN}}_{e}}\right\rangle }_{\mathrm{coll}}=\displaystyle \frac{3}{2}{\sigma }_{T}\;\displaystyle \frac{{m}_{e}{c}^{2}}{{\beta }^{2}}\lambda ,\end{eqnarray} \tag{ 1 }$$

where ${\sigma }_{T}$ is the Thomson cross-section, and

$$\begin{eqnarray}&&\lambda =\mathrm{ln}\left(\displaystyle \frac{{W}_{\mathrm{max}}}{I}\right)\end{eqnarray} \tag{ 2 }$$

is the "Coulomb logarithm." Here W_max is the maximum amount of energy that can be lost by the cosmic ray in a single encounter with an electron and I is the mean excitation energy of the medium associated with individual encounters. In deriving Equation (1), the speeds of the target electrons are assumed negligible compared with that of the cosmic ray. We now proceed to evaluate λ.

The maximum energy transfer is determined by the kinematics of a head-on collision. For low-energy protons (energies $\ll 1\;\mathrm{TeV}$) the kinematic limit is

$$\begin{eqnarray}&&{W}_{\mathrm{max}}\simeq 2\;{\beta }^{2}{\gamma }^{2}\;{m}_{e}{c}^{2}\qquad {\rm{(protons)}},\end{eqnarray} \tag{ 3 }$$

where γ is the cosmic-ray Lorentz factor. For cosmic-ray electrons, on the other hand, one takes

$$\begin{eqnarray}&&{W}_{\mathrm{max}}={\left(\displaystyle \frac{\gamma +1}{2}\right)}^{1/2}(\gamma -1){m}_{e}{c}^{2}\qquad {\rm{(electrons)}},\end{eqnarray} \tag{ 4 }$$

(Jackson 1962) as the maximum energy transfer. For both types of primary, the maximum energy transfer is the same for both neutral and ionized gases.

For neutrals, the mean excitation energy is comparable to the ionization energy of the medium. It is difficult to calculate and in practice it is usually determined by fitting to data. We will consider only the case of a pure hydrogen target, for which the U.S. National Institute of Standards (NIST) recommends

$$\begin{eqnarray}&&I=19.2\;{\rm{eV}}\qquad (\mathrm{neutral}\ \mathrm{hydrogen}),\end{eqnarray} \tag{ 5 }$$

as an appropriate value for both electrons and protons.¹

On the other hand, if all the electrons in the target medium are already free, then the hydrogen ionization energy is no longer relevant. In that case, the mean excitation energy is the quantum of energy associated with Langmuir oscillations in the plasma (Larkin 1959; Ginzburg & Syrovatskii 1964):

$$\begin{eqnarray}&&I={\hslash }{\omega }_{p}\qquad (\mathrm{ionized}\ \mathrm{hydrogen}),\end{eqnarray} \tag{ 6 }$$

with ${\omega }_{p}$ being the plasma frequency. Numerically we have ${\omega }_{p}\simeq 5.6\times {10}^{4}\;\sqrt{{n}_{e}}\ \mathrm{rad}\;{{\rm{s}}}^{-1}$, with n_e in ${\mathrm{cm}}^{-3}$. Thus, if we adopt ${n}_{e}\sim 0.1\;{\mathrm{cm}}^{-3}$  for the WIM (e.g., Haffner et al. 2009), then $I\sim {10}^{-11}\;\mathrm{eV}$. The difference in I values for neutral versus ionized hydrogen is so great that the energy loss rate (1) can be substantially larger for a plasma than for a neutral gas despite the fact that I only appears in the logarithm.

Figure 1 shows the foregoing results in graphical form for cosmic-ray electrons/protons passing through a gas of neutral or ionized hydrogen. Also plotted are values from the NIST databases (see footnote 1) of collisional stopping power for fast particles in neutral hydrogen: pSTAR and eSTAR for protons and electrons, respectively. For LECR protons, the differences between pSTAR values and model (1) are better than 10% for $E\gtrsim 100\;\mathrm{keV}$. At lower energies we enter the regime where the proton speed is $\beta \lesssim \alpha \simeq 1/137$, the speed of electrons bound in hydrogen, so the impulse approximation used to derive Equation (1) is no longer valid. For cosmic-ray electrons this circumstance is not reached until $E\lesssim 100\;\mathrm{eV}$, and there is agreement between the predictions of model (1) and the eSTAR values to better than 5% from $1\;\mathrm{keV}$ to $100\;\mathrm{MeV}$. Because our model works well for neutral gas we expect it to provide a good approximation to the Coulomb losses in ionized gas, providing only that the thermal speeds of the target electrons are negligible. For the WIM, we are interested in plasma temperatures of order ${10}^{4}\;{\rm{K}}$, which limits the applicability of model (1) to protons of energy $\gtrsim 1\;{\rm{keV}}$ and electrons of energy $\gtrsim 1\;{\rm{eV}}$, and thus includes the full range of energies shown in Figure 1.

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In subsequent calculations we use the NIST values of Coulomb losses for neutral hydrogen; for ionized hydrogen we use model (1) with an assumed density of ${n}_{e}=0.1\;{\mathrm{cm}}^{-3}$. We neglect all other loss processes, so our model only applies to LECR ($E\lesssim 100\;\mathrm{MeV}$). We now turn to the calculation of the total Coulomb loss rate for LECR in the solar neighborhood.

It has already been established (Indriolo et al. 2009; Padovani et al. 2009) that flat LECR spectra cannot explain the observed ionization rates of diffuse molecular gas; the spectrum must rise substantially below $100\;{\rm{MeV}}$. In this paper, therefore, we consider only steep LECR spectra, for which the total Coulomb losses are dominated by the low-energy particles in the spectrum. Unfortunately the detailed spectral shape of LECR in the solar neighborhood is not known. We also do not know whether protons or electrons are principally responsible for the ionization of diffuse molecular gas (Padovani et al. 2009). Furthermore, the low-energy spectrum of either species is modified during propagation. We can nevertheless estimate LECR heating in ionized gas simply by scaling from the measured ionization rate in neutral gas as follows.

Equation (1) describes the rate at which a single cosmic ray loses energy via Coulomb collisions. For a particle spectrum j(E) (${\mathrm{cm}}^{-2}\;{{\rm{s}}}^{-1}\;{\mathrm{sr}}^{-1}\;{\mathrm{MeV}}^{-1}$), the total rate at which energy is lost, i.e., the total dissipation rate, H, is just the integral of the stopping power over the spectrum and solid-angle, Ω:

$$\begin{eqnarray}&&H=-\int \int d{\rm{\Omega }}\;{dE}{\left\langle \displaystyle \frac{{dE}}{{{dN}}_{e}}\right\rangle }_{\mathrm{coll}}j(E).\end{eqnarray} \tag{ 7 }$$

Figure 1 shows that except for the lowest energy protons ($E\lt 0.1\;\mathrm{MeV}$) the ratio of stopping powers in neutral and ionized media varies only slowly with particle energy. So if we have a rough estimate of the particle energies which dominate the total dissipation, we can evaluate the ratio of stopping powers in the two media at that energy and that should give us a good estimate of the ratio of the total dissipation rates.

In neutral gas the dissipation rate can be gauged by the primary ionization rate, ζ. Indriolo et al. (2007) summarized data on the ionization rate of diffuse molecular gas in the solar neighborhood and arrived at the conclusion that ${\mathrm{log}}_{10}\zeta ({{\rm{s}}}^{-1}\ {\mathrm{proton}}^{-1})\simeq -15.7\pm 0.5$. For fast electrons in molecular hydrogen, Dalgarno et al. (1999) determined the mean energy required per ion pair to be $\langle {E}_{\mathrm{ion}}\rangle \simeq 40\;\mathrm{eV}$. We adopt this value for cosmic-ray protons also. Thus, in diffuse molecular gas in the solar neighborhood, the LECR dissipation rate is

$$\begin{eqnarray}&&{\mathrm{log}}_{10}H={\mathrm{log}}_{10}\zeta +{\mathrm{log}}_{10}\langle {E}_{\mathrm{ion}}\rangle =-25.9\pm 0.5,\end{eqnarray} \tag{ 8 }$$

with H in units of $\mathrm{erg}\;{{\rm{s}}}^{-1}\;{\mathrm{proton}}^{-1}$.

The clouds considered by Indriolo et al. (2007) have a broad range of column densities and sizes; we adopt values of ${N}_{e}\sim 3\times {10}^{21}\;{\mathrm{cm}}^{-2}$ and $L\sim 4\;{\rm{pc}}$, respectively, as being representative. As a result of scattering by magnetic irregularities, cosmic-ray transport is diffusive and a cosmic ray that traverses an entire cloud may encounter a total column density much larger than the line-of-sight value given by Indriolo et al. (2007). There is currently insufficient data on LECR to permit useful constraints on the quantitative characteristics of their diffusion. For GeV cosmic rays, values of the diffusion coefficient of $D\sim 4\times {10}^{28}\;{\mathrm{cm}}^{2}\;{{\rm{s}}}^{-1}$ typically provide a good match to the data (Strong et al. 2007) and correspond to a mean free path to large-angle scattering of $3D/c\sim 1\;{\rm{pc}}$. We assume that this mean free path also holds for LECR, implying that the column density encountered by a cosmic ray traversing a typical cloud in the sample of Indriolo et al. (2007) is ${N}_{e}\sim {10}^{22}\;{\mathrm{cm}}^{-2}$. Because the LECR spectrum is steep, the total dissipation rate in a typical cloud should therefore be dominated by particles whose range is $\sim {10}^{22}\;{\mathrm{cm}}^{-2}$.

Figure 2 shows the ranges of electrons and protons in neutral hydrogen determined by integrating the reciprocal of the stopping powers shown in Figure 1. From Figure 2 we can see that particle ranges $\sim {10}^{22}\;{\mathrm{cm}}^{-2}$ correspond to protons of energy $\sim 5\;\mathrm{MeV}$ and electrons of energy $\sim 0.2\;\mathrm{MeV}$. It is particles of about these energies that we expect to make the main contribution to the ionization rates given by Indriolo et al. (2007). As can be seen in Figure 1, such particles have stopping powers which are respectively 5.4 and 4.1 times greater in the WIM than in neutral gas. See also Figure 3 where the ratios of stopping power are plotted as a function of particle range in neutral gas.² In the case of ionized gas all of the power dissipated from LECR goes into heat, so the estimated heating rate of the WIM in the solar neighborhood (in $\mathrm{erg}\;{{\rm{s}}}^{-1}\;{\mathrm{proton}}^{-1}$) is:

$$\begin{eqnarray}&&{\mathrm{log}}_{10}{H}_{p}\simeq -25.2\pm 0.5\end{eqnarray} \tag{ 9 }$$

if the LECR are predominantly protons, and

$$\begin{eqnarray}&&{\mathrm{log}}_{10}{H}_{e}\simeq -25.3\pm 0.5\end{eqnarray} \tag{ 10 }$$

if the LECR are predominantly electrons.

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The midpoints of estimates (9) and (10) are both below the heating rate needed to maintain the WIM at ${10}^{4}\;{\rm{K}}$, which is ${\mathrm{log}}_{10}{H}_{\mathrm{WIM}}\simeq -25.1$ at a density of ${n}_{e}=0.1\;{\mathrm{cm}}^{-3}$ (Reynolds 1990). However, neither estimate is far below the requisite power. Even at the lower end of the indicative range, LECR are expected to contribute 20%–25% of the WIM's power. At the upper end of the range the WIM can be maintained against radiative cooling by LECR heating alone, regardless of the nature of the primaries.

The estimates (9) and (10) are valid for the WIM in the neighborhood of the Sun, with "neighborhood" defined by the sample of diffuse molecular clouds in Indriolo et al. (2007). The distances of the target stars in that sample range up to a few ${\rm{kpc}}$, but most are below $1\;{\rm{kpc}}$ so we expect the intervening diffuse clouds to lie at distances of order a few hundred parsec from the Sun. As the target stars lie at low Galactic latitudes, this distance estimate is in effect a radius within the Galactic plane. The estimates (9) and (10) should, however, be valid at least as far off the plane as LECR can diffuse. If, as assumed above, the mean free path to the scattering of LECR is $\sim 1\;{\rm{pc}}$, then electrons of energy $\sim 0.2\;\mathrm{MeV}$ and protons of energy $\sim 5\;\mathrm{MeV}$ should be able to diffuse a distance $\sim {10}^{2}\;{\rm{pc}}$ through the WIM. Absent a reliable model of acceleration and propagation for LECR, we cannot say whether  estimates (9) and (10) are appropriate at larger distances from the plane.

3.1. The Influence of Helium

Strong LECR fluxes have been inferred in a number of studies of the hard X-ray emission of the inner Galactic disk. Although X-ray emission can arise from either electrons (bremßtrahlung) or protons (inverse-bremßtrahlung), models in which a large fraction of the hard X-ray emission is attributed to LECR ions are disfavored because they tend to predict too much nuclear gamma-ray line emission from the ISM and too much beryllium production via spallation (Tatischeff et al. 1999; Valinia et al. 2000). At present there is no consensus on the shape of the LECR electron spectrum; nevertheless, high ionization and dissipation rates are a common feature of the models (Skibo & Ramaty 1993; Skibo et al. 1996; Valinia & Marshall 1998; Valinia et al. 2000).

In some cases authors quote the ionization rate implied by their models. For example, Skibo & Ramaty (1993) modeled the 0.03–1000 MeV hard X-ray/gamma-ray data from the central radian of the Galaxy and estimated a primary ionization rate of $\zeta \sim {10}^{-15}\;{{\rm{s}}}^{-1}$ due to the LECR electrons in their model. This estimate is an average over the inner Galactic disk, so it applies to the ISM throughout that region. It is five times larger than the local value of ζ from Indriolo et al. (2007) used in Section 3. If the shape of the LECR spectrum is similar to that local to the Sun, the implied heating rate for the WIM is then five times larger than the estimate given in Equation (10).

The high ionization rate deduced by Skibo & Ramaty (1993) is associated with the low-energy end of their model electron spectrum, where fluxes must be large if the $30\;\mathrm{keV}$ X-rays are primarily from bremßtrahlung. Skibo et al. (1996) argued in favor of a large bremßtrahlung contribution to the diffuse Galactic emission down to $\sim 10\;\mathrm{keV}$, which pushes the LECR ionization rate even higher. They inferred a primary ionization rate of $1.6\times {10}^{-14}\;{{\rm{s}}}^{-1}$, suggesting a heating rate that is 80 times larger than the estimate in Equation (10).

Decreasing the contribution made by LECR electrons to the observed $10\;\mathrm{keV}$ emission naturally leads to lower estimates of the dissipation rate. Valinia et al. (2000) modeled the $\lt 10\;\mathrm{keV}$ X-ray spectrum of the inner Galaxy with a combination of thermal plasma and non-thermal electrons. They did not quote an ionization rate, but they did give their model LECR electron spectrum:

$$\begin{eqnarray}&&{J}_{e}=1.7\times {10}^{-6}{\left(\displaystyle \frac{E}{{E}_{0}}\right)}^{0.3}\mathrm{exp}(-E{/E}_{0}),\end{eqnarray} \tag{ 11 }$$

in ${\mathrm{cm}}^{-3}\;{\mathrm{sr}}^{-1}\;{\mathrm{MeV}}^{-1}$, with ${E}_{0}=0.09\;\mathrm{MeV}$. The corresponding heating rate $(\mathrm{erg}\ {{\rm{s}}}^{-1}\;{\mathrm{proton}}^{-1})$ for the WIM can be evaluated from Equation (7), wth ${j}_{e}={J}_{e}\;\beta c$, it is:

$$\begin{eqnarray}&&{\mathrm{log}}_{10}{H}_{e}\simeq -23.5,\end{eqnarray} \tag{ 12 }$$

i.e., roughly 60 times larger than the local estimate (10).

The approach of Valinia et al. (2000) simultaneously modeled the properties of the Fe K emission line, which exhibits a fluorescent component at $6.4\;\mathrm{keV}$, and the X-ray continuum shape. A recent study of hard X-ray line and continuum emission from the inner few degrees of the Galactic disk concluded that LECR protons explain the data better than LECR electrons (Nobukawa et al. 2015). Their preferred model proton spectrum is steep and has a large energy density, and the implied heating rate of the WIM is three orders of magnitude larger than our local estimate (9). That is so large that it may be difficult to identify a power source able to maintain such a spectrum in steady state. For the present discussion, the important point is that an explanation of the hard X-rays that relies on LECR protons rather than electrons does not evade the implication of a large heating rate for the WIM.

We can summarize this section in the following way: if a significant fraction of the observed hard X-ray emission from the inner Galaxy is due to LECR, then those LECR heat the WIM at a much greater rate than the local estimates given in Section 3.

Various processes have been contemplated as heat sources for the WIM, including photoionization (e.g., Mathis 1986); photoelectric heating by dust grains (Reynolds & Cox 1992; Weingartner & Draine 2001); dissipation of hydromagnetic wave energy (Minter & Spangler 1997; Wiener et al. 2013)³ ; and, of course, the Coulomb collisions of cosmic rays. Of these, photoionization and LECR heating are of special interest because they may also be able to account for the ionization of the WIM. Lyman continuum photons and LECR are similar in that both appear able to supply the necessary power, and ultimately they have a common origin in young, massive stars (which are the progenitors of supernovae). But LECR and UV photons differ profoundly in the way they propagate through the Galaxy. The large absorption cross-section of neutral hydrogen to the Lyman continuum ($\sim {10}^{-18}\;{\mathrm{cm}}^{2}$) means that these UV photons are not expected to travel far from their source. By contrast cosmic rays are able to penetrate much larger columns of gas, particularly neutral gas, depending on their energy (see Figure 2). Because of this difference, LECR are more readily able to supply the widespread power needed to explain the WIM.

Many authors have argued that non-uniform gas density in the ISM can lead to Lyman continuum photons propagating to large distances from the source in some directions (e.g., Dove & Shull 1994; Wood et al. 2010), making it possible to explain the broad spatial distribution of the WIM in terms of photoionization. However, the spectrum of the WIM is also a problem for photoionization models: the observed forbidden-line ratios are distinctly different from those exhibited by "classical" H ii regions (which are certainly photoionized), consistent with the WIM being significantly hotter (Haffner et al. 1999). Moreover, the observed trends in line ratios with gas density suggest that WIM heating cannot be solely the heating associated with ionization (Haffner et al. 1999; Reynolds et al. 1999). Because LECR heat the free electrons themselves as well as contributing heat by ionizing neutrals, the observed forbidden-line ratios pose no special problem for models in which LECR dominate the heating of the WIM.

The potential importance of LECR in determining the state of interstellar gas has long been appreciated. Indeed, theoretical models in which LECR are the principal source of heat and ionization actually predate the discovery of the WIM (Spitzer & Tomasko 1968; Field et al. 1969; Goldsmith et al. 1969). The main difficulty with such models has always been the very large uncertainty in LECR flux. Substantial uncertainty remains and there are indications that LECR fluxes vary substantially from place to place, even in the solar neighborhood (Indriolo & McCall 2012), but it now seems likely that LECR are playing an important role in the WIM and that detailed models specific to that context would be valuable. In addition to the key issue of whether one can build an acceptable model of the WIM based on LECR heating and ionization, there is the prospect that the available nebular diagnostics for the WIM could provide some new insights into the LECR population.

Assuming Case B conditions, with a temperature of ${10}^{4}\;{\rm{K}}$, and a density of ${n}_{e}\sim 0.1\;{\mathrm{cm}}^{-3}$, the recombination rate in the WIM is $\sim 3\times {10}^{-15}\;{\mathrm{cm}}^{-3}\;{{\rm{s}}}^{-1}$. Reynolds et al. (1998) showed that [O i]$\ \lambda 6300$ emission from the WIM is weak and inferred that the density of atomic hydrogen in the WIM, n_a, must be small compared to that of ionized hydrogen, so ${n}_{a}\lesssim 0.1\;{\mathrm{cm}}^{-3}$. Accounting for secondary ionizations (e.g., Cravens & Dalgarno 1978), the total LECR ionization rate is approximately $1.7\;{n}_{a}\zeta$, so for this rate to balance recombinations we must have $\zeta \gtrsim 2\times {10}^{-14}\;{{\rm{s}}}^{-1}$, i.e., at least as large as the value inferred by Skibo et al. (1996) for LECR electrons in the inner Galaxy (Section 4). This result is not unreasonable. As already noted, LECR fluxes may vary significantly from place to place in the Galaxy, and those regions where the ionized fraction is highest are a priori likely to be those where the ionizing fluxes are greatest.

The measured ionization rates in diffuse molecular clouds local to the Sun imply that LECR fluxes are high—high enough that they may be able to maintain the WIM at ${10}^{4}\;{\rm{K}}$ without the need for any other source of heat. LECR also lead to diffuse, hard X-ray emission from the ISM, and X-ray observations of the inner Galactic disk provide independent evidence of strong LECR fluxes. The WIM heating rates implied by the X-ray data are model-dependent, but they're consistently even higher than our local estimate from the molecular ion data. Although large amounts of power flow in both Lyman continuum photons and LECR, the spatial distribution and spectral properties of the WIM favor LECR as a more natural explanation of the requisite heating.

Thanks to Tony Bell, Don Melrose, and Mark Wardle for their help.