ELECTRON IMPACT DISSOCIATION X ¹Σ⁺_g → b ³Σ_u⁺ AND EXCITATIONS X ¹Σ⁺_g → a ³Σ_g⁺ AND X ¹Σ⁺_g → B ¹Σ_u⁺ OF MOLECULAR HYDROGEN IN NONTHERMAL ASTROPHYSICAL PLASMAS

Electron–atom and electron–molecule collisions are the most fundamental physical processes in weakly ionized astrophysical plasmas (Dalgarno & Layzer 1987; Cox 2000; Itikawa 2007). In particular, the electronic excitation transitions of molecules in astrophysical plasmas have been of great interest since line emissions due to molecular electronic transitions from excited states provide useful spectral information on the physical properties of the surrounding astrophysical plasma (Tawara et al. 1990; Yoon et al. 2008). It has been well known that molecular hydrogen is the most abundant molecule in the interstellar medium, intergalactic medium, and also in planetary atmospheres (Gould & Salpeter 1963; Gould et al. 1963; Krishna Swamy 1975; Glover & Brand 2003; Draine 2011). Inelastic electron–molecular-hydrogen collisions played an important role in the formation of astrophysical phenomena such as the interstellar medium (Khristenko et al. 1998; Shevelko & Tawara 1998, 2012; Rau 2002; Ki & Jung 2012) because they profoundly contributed to the energy loss of plasma electrons in such environments. A detailed discussion of the formation of the H₂ molecule and the dominant reaction converting ortho- to para-H₂ due to proton exchange can be found in the excellent work of Fower (1990). Another detailed discussion of the dissociate process of vibrationally excited H₂, H₂(X, v) → H*₂(B ¹Σ⁺_u) → H + H, can also be found in Stibbe & Tennyson (1999). Furthermore, it is shown that the reaction H⁻ + H → H₂ + e⁻ is the dominant channel for forming H₂ in the absence of dust grains in the interstellar medium (McDowell 1962). However, in regions of the interstellar medium containing dust grains, it has been known that the H₂ molecule is formed mainly via catalysis on the surface of dust grains, i.e., grain catalysis of H₂ (Gould & Salpeter 1963; Gould et al. 1963; Hollenbach & Salpeter 1970, 1971; Osterbrock & Ferland 2006). A detailed discussion of the total grain geometric cross section per H nucleon has been obtained by Weingartner & Drain (2001) using the silicate–graphite–polycyclic-aromatic-hydrocarbon (PAH) grain model. Hence, the influence of the charge of dust grains on the formation of H₂ molecules and the total grain geometric cross section by catalysis on the surface of dust grains will be treated elsewhere. Detailed theoretical calculations have provided the mean energies and yields for the ionization, electronic excitation, dissociation, and vibrational excitation produced by energetic electrons traversing a gas mixture containing molecular hydrogen (Dalgarno et al. 1999). The Jet Propulsion Laboratory group of Ajello has contributed a quite useful and considerable amount of data for molecular hydrogen from their systematic studies of electron-impact UV emissions (Ajello & Shemansky 1993; Liu et al. 1995, 1998, 2003; James et al. 1998; Jonin et al. 2000). For molecular hydrogen, the three electronic excitation transitions: X ¹Σ⁺_g → b ³Σ_u⁺, X ¹Σ⁺_g → a ³Σ_g⁺, and X ¹Σ⁺_g → B ¹Σ_u⁺ provide the main contributions to the electric excitation processes via electron impacts (Tawara et al. 1990; Itikawa 2007; Yoon et al. 2008). The Yukawa-type screened Coulomb interaction in thermal astrophysical plasmas has also been investigated by using the standard Debye–Hückel model acquired by the Maxwellian velocity distribution because the interaction energy in the plasma is usually smaller than the plasma kinetic energy and there is no energy exchange between the plasma and the surrounding environment of the thermal plasma (Shevelko & Vainshtein 1993; Fujimoto 2004; Smirnov 2007). In astrophysical plasmas, nonthermal velocity distributions are ubiquitous as the nonthermal distribution function, including high-velocity tails, is readily generated by the interaction between external perturbation and the surrounding plasma particles. In the presence of an external radiation field in astrophysical plasmas, it has been shown that the deviated Maxwellian velocity distribution would be represented by a nonthermal Lorentzian velocity distribution with a spectral index and characteristic temperature of Lorentzian plasma (Hasegawa et al. 1985; Scudder 1992; Mendis & Rosenberg 1994; Baumjohann & Treumann 1996; Rubab & Murtaza 2006; Jung 2009). Hence, it can be expected that the electronic excitations of molecular hydrogen in thermal Maxwellian astrophysical plasmas would be quite different from those in nonthermal astrophysical Lorentzian plasmas due to the influence of the spectral index and the effective Debye distance in Lorentzian plasmas. Thus, in this paper, we have investigated the properties of the electronic excitations and the influence of the nonthermal character of the Lorentzian plasma distribution on the X ¹Σ⁺_g → b ³Σ_u⁺, X ¹Σ⁺_g → a ³Σ_g⁺, and X ¹Σ⁺_g → B ¹Σ_u⁺ electronic excitation transitions of molecular hydrogen by studying electron impacts in astrophysical Lorentzian plasmas.

In Section 2, we discuss the astrophysical Lorentzian distribution of plasma electrons and the effective Debye distance in nonthermal astrophysical plasmas. In Section 3, we discuss the X ¹Σ⁺_g → b ³Σ_u⁺, X ¹Σ⁺_g → a ³Σ_g⁺, and X ¹Σ⁺_g → B ¹Σ_u⁺ electronic excitation processes of H₂ by electron impact. We also obtain the analytic forms of the X ¹Σ⁺_g → b ³Σ_u⁺, X ¹Σ⁺_g → a ³Σ_g⁺, and X ¹Σ⁺_g → B ¹Σ_u⁺ electronic excitation cross sections as functions of the collision energy. In Section 4, we obtain the closed forms of the X ¹Σ⁺_g → b ³Σ_u⁺, X ¹Σ⁺_g → a ³Σ_g⁺, and X ¹Σ⁺_g → B ¹Σ_u⁺ electronic excitation rates of molecular hydrogen by the nonthermal Lorentzian distribution of plasma electrons as functions of the spectral index and characteristic temperature of the Lorentzian plasma. We also discuss nonthermal effects on the X ¹Σ⁺_g → b ³Σ_u⁺, X ¹Σ⁺_g → a ³Σ_g⁺, and X ¹Σ⁺_g → B ¹Σ_u⁺ electronic excitation rates of molecular hydrogen in astrophysical Lorentzian plasmas. Finally, the summary is given in Section 5.

In thermal equilibrium astrophysical plasmas, usually there is no energy exchange between the plasma and the external environment. However, in most astrophysical circumstances, it has been found that deviations from the conventional thermal Maxwellian velocity distribution would be expected as a result of interaction between external perturbations and the surrounding plasma particles. A pioneering study by Hasegawa et al. (1985) has shown that in the presence of an external radiation field, the deviated Maxwellian velocity distribution would be in the form of the nonthermal Lorentzian velocity distribution. A careful and detailed investigation of the spectral properties of the Lorentzian distribution was performed by Summers & Thorne (1992). Moreover, it has been shown that the nonthermal plasma distribution function usually encompasses a nonthermal high-velocity part that is formed by the Lorentzian (kappa) distribution function (Hasegawa & Sato 1989; Scudder 1992; Mendis & Rosenberg 1994). These Lorentzian velocity distributions with departed high-velocity tails are frequently found in the step temperature gradient regions in astrophysical nonthermal plasmas. Hence, the nonthermal astrophysical plasma accommodating the high-velocity tail for the plasma distribution function can be expressed by the following form of the Lorentzian distribution function f_κ(v):

$$\begin{eqnarray} f_\kappa (v)dv &=& 4\pi \left({\frac{m}{{2\pi \kappa E_\kappa }}} \right)^{3/2} \frac{{\Gamma (\kappa + 1)}}{{\Gamma (\kappa - 1/2)}}v^2\nonumber\\ &&\times \left({1 + \frac{{mv^2 }}{{2\kappa E_\kappa }}} \right)^{ - (\kappa + 1)} dv, \end{eqnarray} \tag{ 1 }$$

where κ is the spectral index of the Lorentzian plasma, m is the mass of the electron, E_κ[ = E_M(κ − 3/2)/κ] is the characteristic Lorentzian energy, E_M(= k_BT) is the Maxwellian energy, k_B is the Boltzmann constant, T is the plasma temperature, and Γ(x) represents the Euler gamma function with argument x. In addition, it has been found that radiation perturbation modifies the diffusion process in astrophysical plasmas since the non-Columbic diffusion coefficient can be induced through interaction with the external radiation field and is also found to be proportional to the square of the velocity v of the plasma particle (Hasegawa & Sato 1989). Accordingly, it was evident that the modification of the total diffusion coefficient in astrophysical plasmas would be obtained by the factor (1 + α_rv²), where α_r is a constant related to the intensity of the radiation field (Hasegawa et al. 1985; Hasegawa & Sato 1989). It is also interesting to recognize that the nonthermal Lorentzian (kappa) velocity distribution function contains important physical properties. First, the Lorentzian velocity distribution takes the form of a simple power-law distribution function at high energies such as f_κ(v)∝v²(mv²/2κE_κ)^{−(κ + 1)}. Therefore, nonthermal Lorentzian distributions become power-law distributions when mv²/2 ≫ κE_κ. Moreover, the Lorentzian velocity distribution turns out to be the standard Maxwellian distribution for all energies in the absence of external perturbation, i.e., when the special index reaches infinity, such as $f_{\rm M} (v) \propto v^2 e^{ - mv_{}^2 /2E_{\rm M} }$, due to the limiting relation $\stackrel{\lim} {_{n \to \infty} } (1 + a/n)^n = e^a$ (Arfken & Weber 2005). We have then found that the nonthermal Lorentzian velocity distribution f_κ(v) encompasses a very wide range of plasma velocity distributions from the standard Maxwellian distribution to the inverse power-law distribution. Furthermore, it has been found that the effective Debye shielding distance λ_κ in nonthermal Lorentzian plasmas can be obtained by λ_κ = λ_Dμ^1/2_κ (Rubab & Murtaza 2006; Ki & Jung 2012), where λ_D is the conventional Debye shielding distance in thermal Maxwellian plasmas and μ_κ[ ≡ (κ − 3/2)/(κ − 1/2)] represents the fractional measure of the nonthermal population in astrophysical Lorentzian plasmas due to the nonthermal character of the plasma. The solar coronal plasmas have been determined to be parts of the nonthermal velocity tail portions of Lorentzian distributions with spectral indices 2.62 < κ < 3.15 for electrons and 2.06 < κ < 2.69 for protons (Parks 2004). Hence, we shall investigate the X ¹Σ⁺_g → b ³Σ_u⁺, X ¹Σ⁺_g → a ³Σ_g⁺, and X ¹Σ⁺_g → B ¹Σ_u⁺ electronic excitation rates of molecular hydrogen in astrophysical Lorentzian plasmas with the distribution function f_κ(v).

In astrophysics, the cross section data for collision and radiation processes involving molecules are necessary to understand the behavior of the astrophysical environment. It has been known that the available data on electron impact excitation cross sections for electronic states in molecular hydrogen (H₂) obtained by electron-scattering techniques are quite limited due to the difficulty of resolving the overlapping band structure of electronic transitions in the electron energy loss spectra (Khristenko et al. 1998). It has also been shown that the transition between electronic states of a diatomic molecule is accompanied by a transition of rotational and vibrational states (Itikawa 2007). It is widely known that the radiation field modifies the diffusion coefficient in plasmas since the radiation field causes non-Columbic diffusion which is proportional to the square of the plasma velocity (Hasegawa et al. 1985). Under these circumstances, the electron distribution would be found in the nonthermal Lorentzian form. Hence, in this work we assume that molecular hydrogen resides in the ground vibrational level and the lowest few rotational levels such as the J = 0 and J = 1 rotational levels in low-density astrophysical plasmas with a Lorentzian distribution of electrons. A detailed investigation on the J = 0 → 2 and J = 1 → 3 rotational excitation of the molecular hydrogen in astrophysical Lorentzian plasmas was given by Ki & Jung (2012). For the X ¹Σ⁺_g → b ³Σ_u⁺ electronic excitation, the state b ³Σ⁺_u is known as a repulsive state where molecular hydrogen promptly dissociates into two hydrogen atoms in the ground state (Sharp 1971). The direct excitation of the b ³Σ⁺_u anti-bonding state is the dominant process leading to the dissociation of molecular hydrogen (Khristenko et al. 1998). The potential energies for the ground state and excited states of molecular hydrogen have been obtained from quantum chemistry calculations by Kolos & Wolniewicz (1965, 1968, 1969) even though the experimental date is somewhat rare. In the R-matrix method, the coordinate space of the scattered electron would be divided into the internal region for the short-range part of the interaction and the external region for the long-range part of the interaction with a sphere radius as the boundary between these two regions (Khristenko et al. 1998). The reliable X ¹Σ⁺_g → b ³Σ_u⁺ molecular data have been given for the energies up to 20 eV (Khakoo et al. 1987) and for 30–100 eV (Khakoo & Segura 1994), respectively. By using these molecular data, the accurate and useful fitting formula for the X ¹Σ⁺_g → b ³Σ_u⁺ electronic excitation cross section $\sigma _{b\,{}^3\Sigma _u^ + } (E^\prime)$ is then found to be

$$\begin{equation} \sigma _{b\,{}^3\Sigma _u^ + } (E^{\prime}) = \sigma _0 \frac{{A_0 + A_1 E^{\prime} + A_2 E^{\prime2} }}{{1 + A_3 E^{\prime} + A_4 E^{\prime2} + A_5 E^{\prime3} }}, \end{equation} \tag{ 2 }$$

where σ₀ ≡ 10⁻¹⁶ cm², the coefficients A_j are given in Table 1, E' ≡ E/1 eV, E(≡ m_ev²/2) is the kinetic energy of the incident electron, m_e is the mass of the electron, and v is the velocity of the incident electron. This fitting formula is reliable for 9.2 eV < E < 100 eV. In this work, the accuracies for the fitting formulae are obtained to be better than 90% of the accuracy compared with the current accurate numerical molecular data surveyed through the end of 2006 (Yoon et al. 2008). The peak energy of the X ¹Σ⁺_g → b ³Σ_u⁺ electronic excitation cross section is known to be at about 17 eV (Yoon et al. 2008). Additionally, the differential electronic excitation cross section of the X ¹Σ⁺_g → b ³Σ_u⁺ transition has been obtained as a function of the scattering angle (Khristenko et al. 1998) and the peak scattering angles of the differential cross sections for the incident energies E = 25 eV and 50 eV are, respectively, found to be $\theta [\deg ] \approx 45$ and 35 due to the preference of forward scattering for higher incident electron energy. For the X ¹Σ⁺_g → a ³Σ_g⁺ electronic excitation, the excitation energy to the a ³Σ⁺_g state has been acknowledged as 11.79 eV through experimental measurement. The potential energy curve for the a ³Σ⁺_g state of molecular hydrogen has been obtained from quantum chemistry calculations by Kolos & Wolniewicz (1968). In addition, the peak energy of the X ¹Σ⁺_g → a ³Σ_g⁺ electronic excitation cross section is known to be at around 17 eV (Yoon et al. 2008). Reliable X ¹Σ⁺_g → a ³Σ_g⁺ molecular data have been given for energies up to 30 eV (Khakoo & Trajmar 1986) and above 30 eV (Wrkich et al. 2002). By using these data, the accurate fitting coefficients in Equation (2) for the X ¹Σ⁺_g → a ³Σ_g⁺ electronic excitation cross section $\sigma _{a\,{}^3\Sigma _g^ + } (E^\prime)$ are given in Table 1. This fitting formula would be reliable for 11.8 eV < E < 200 eV. For the X ¹Σ⁺_g → B ¹Σ_u⁺ electronic excitation, the excitation energy to the B ¹Σ⁺_u state has been determined to be 11.18 eV by experimental measurement (Yoon et al. 2008). Reliable X ¹Σ⁺_g → B ¹Σ_u⁺ molecular data have been obtained through the measurement of electron energy loss spectra (Wrkich et al. 2002). By using these data, the accurate fitting coefficients in Equation (2) for the X ¹Σ⁺_g → B ¹Σ_u⁺ electronic excitation cross section $\sigma _{B\,{}^1\Sigma _u^ + } (E^\prime)$ are also given in Table 1. This fitting formula would be reliable for 12.0 eV < E < 3000 eV. The peak of the X ¹Σ⁺_g → B ¹Σ_u⁺ electronic excitation cross section is at about 45 eV (Yoon et al. 2008). The fitting coefficients for the electron impact dissociation and excitation cross sections $\sigma _{b\,{}^3\Sigma _u^ + }$, $\sigma _{a\,{}^3\Sigma _g^ + }$, and $\sigma _{B\,{}^1\Sigma _u^ + }$ are given in Table 1. The fitting formula and coefficients (Table 1) for the X ¹Σ⁺_g → b ³Σ_u⁺ excitation cross section $\sigma _{b\,{}^3\Sigma _u^ + } (E^\prime)$ have been obtained by comparing the experimental data (Khakoo et al. 1987; Khakoo & Segura 1994) and compiled data surveyed through the end of 2006 (Yoon et al. 2008). The fitting coefficients (Table 1) for the X ¹Σ⁺_g → a ³Σ_g⁺ excitation cross section $\sigma _{a\,{}^3\Sigma _g^ + } (E^\prime)$ are obtained by comparing the experimental data (Khakoo & Trajmar 1986; Ajello & Shemansky 1993; Wrkich et al. 2002) and compiled data (Yoon et al. 2008). In addition, the fitting coefficients (Table 1) for the X ¹Σ⁺_g → B ¹Σ_u⁺ excitation cross section $\sigma _{B\,{}^1\Sigma _u^ + } (E^\prime)$ are obtained by comparing the experimental (Liu et al. 1998) and compiled data (Yoon et al. 2008). The electronic excitation rates R_ex of H₂ by the electron impact in a plasma with the plasma velocity distribution function f(v) would then be represented by

$$\begin{eqnarray} R_{\rm ex} &=& \langle {v\sigma _{\rm ex} (v)} \rangle _{v \ge v_{\rm ex}^{\rm th} } \nonumber \\ &=&\int_{v \ge v_{\rm ex}^{\rm th}} {v\sigma _{\rm ex} (v)f(v)} dv, \end{eqnarray} \tag{ 3 }$$

where the bracket notation 〈〉 represents the average with the plasma velocity distribution function, and v^th_ex represents the threshold velocity of the projectile electron for the electronic excitation of H₂. As we can expect from Equation (3), the nonthermal property of the electronic excitation rate of the molecular hydrogen would be classified by the spectral index and temperature of the Lorentzian plasma.

Table 1. The Electronic Impact Dissociation and Excitation Cross Sections ${\sigma _{b{}^3\Sigma _u^ + }}(E^{\prime })$, ${\sigma _{a{}^3\Sigma _g^ + }}(E^{\prime })$, and ${\sigma _{B{}^1\Sigma _u^ + }}(E^{\prime })$ $(\sigma (E^{\prime }) = {\sigma _0}\frac{{{A_0} + {A_1}E^{\prime } + {A_2}{{E^{\prime }}^2}}}{{1 + {A_3}E^{\prime } + {A_4}{{E^{\prime }}^2} + {A_5}{{E^{\prime }}^3}}}({\sigma _0} \equiv {10^{ - 16}}{\rm {c}}{{\rm {m}}^2},E^{\prime } \equiv E/1{\rm {eV}}))$

Coefficients	X1Σ+g → b3Σu+	X1Σ+g → a3Σg+	X1Σ+g → B1Σu+
	${\sigma _{b{}^3\Sigma _u^ + }}(E^{\prime })$	${\sigma _{a{}^3\Sigma _g^ + }}(E^{\prime })$	${\sigma _{B{}^1\Sigma _u^ + }}(E^{\prime })$
	Emin  = 9.2 eV	Emin  = 11.8 eV	Emin  = 12.0 eV
	Emax  = 100 eV	Emax  = 200 eV	Emax  = 3000 eV
A0	47.6640	−444.0450	−0.3870
A1	−10.0740	31.4280	0.00150
A2	0.6511	0.4726	0.0014
A3	55.6890	856.0040	−0.00229
A4	−7.7384	−113.4070	0.0042
A5	0.2384	3.9964	0.00002

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From Equations (1)–(3), the closed form of the X ¹Σ⁺_g → b ³Σ_u⁺ electronic excitation rate coefficient $R_{b\,{}^3\Sigma _u^ + } (\kappa ,\bar {T})$ of H₂ by the electron impact in astrophysical Lorentzian plasmas is obtained by the following form:

$$\begin{eqnarray} R_{b\,{}^3\Sigma _u^ + } (\kappa ,\bar {T}) &=&\langle {v\,\sigma _{b\,{}^3\Sigma _u^ + } (v)} \rangle _{v \ge v_{b\,{}^3\Sigma _u^ + }^{\rm th} } \nonumber\\ &=&R_0 \left({\frac{1}{{\kappa \bar {E}_\kappa }}} \right)^{3/2} \frac{{\Gamma (\kappa + 1)}}{{\Gamma (\kappa - 1/2)}}\nonumber \\ &&\times \int_{\bar {E}_{b\,{}^3\Sigma _u^ + }^{\rm th}}^{\infty } d\bar {E}\frac{{A_0 \bar {E} + \bar A_1 \bar {E}^2 + \bar A_2 \bar {E}^3 }}{{1 + \bar A_3 \bar {E} + \bar A_4 \bar {E}^2 + \bar A_5 \bar {E}^3 }}\nonumber \\ &&\times\left({1 + \frac{{\bar {E}}}{{\bar {E}_\kappa }}} \right)^{ - \kappa - 1} , \end{eqnarray} \tag{ 4 }$$

where $\bar {T} \equiv k_B T/Ry$, Ry(= me⁴/2ℏ² ≈ 13.6 eV) is the Rydberg constant, $R_0 [ {\equiv} \sigma _0 (8Ry/m\pi)^{1/2} ] \approx 2.4541 \times 10^{ - 8} \,{\rm cm}^3 {/}\sec$, $\bar {E}_{b\,{}^3\Sigma _u^ + }^{\rm th} [ {\equiv} E_{b\,{}^3\Sigma _u^ + }^{\rm th} /Ry] = m(v_{b\,{}^3\Sigma _u^ + }^{\rm th})^2 /2Ry$, $\bar {E}_\kappa \equiv E_\kappa /Ry$, $\bar {E}( {\equiv} E/Ry)$ is the scaled kinetic energy of the electron, $\bar A_1 \to \bar a_1 = - 137.0064$, $\bar A_2 \to \bar a_2 = 120.4275$, $\bar A_3 \to \bar a_3 = 757.3704$, $\bar A_4 \to \bar a_4 = - 1431.2945$, and $\bar A_5 \to \bar a_5 = 712.8802$. For the reason that the scaled electron kinetic energy $\bar {E}$ is in units of Ry, the relation between the coefficients A_j and $\bar A_j$ is given by $\bar A_j = A_j \times (13.6)^j$, where j is an integer. In Equation (4), the threshold energy $E_{b\,{}^3\Sigma _u^ + }^{\rm th}$ is given by 10.45 eV as the experimental threshold energy is rare (Yoon et al. 2008). From Equations (1)–(3), the fitting coefficients for the X ¹Σ⁺_g → a ³Σ_g⁺ electronic excitation rate coefficient $R_{a\,{}^3\Sigma _g^ + } (\kappa ,\bar {T})$ of H₂ by the electron impact in astrophysical Lorentzian plasmas are given by $\bar A_1 \to \bar b_1 = 427.4208$, $\bar A_2 \to \bar b_2 = 87.4121$, $\bar A_3 \to \bar b_3 = 11641.6544$, $\bar A_4 \to \bar b_4 = - 20975.7587$, and $\bar A_5 \to \bar b_5 = 10052.7684$. The excitation threshold $E_{a\,{}^3\Sigma _g^ + }^{\rm th}$ for the X ¹Σ⁺_g → a ³Σ_g⁺ transition is given by 12.41 eV (Yoon et al. 2008). In addition, from Equations (1)–(3), the fitting coefficients for the X ¹Σ⁺_g → B ¹Σ_u⁺ electronic excitation rate coefficient $R_{B\,{}^1\Sigma _u^ + } (\kappa ,\bar {T})$ of H₂ by the electron impact in astrophysical Lorentzian plasmas are found to be $\bar A_1 \to \bar c_1 = 0.2040$, $\bar A_2 \to \bar c_2 = 0.2589$, $\bar A_3 \to \bar c_3 = - 0.3114$, $\bar A_4 \to \bar c_4 = 0.7768$, and $\bar A_5 \to \bar c_5 = 0.0503$. The excitation threshold $E_{B\,{}^1\Sigma _u^ + }^{\rm th}$ for the X ¹Σ⁺_g → B ¹Σ_u⁺ transition is given by 13.15 eV (Yoon et al. 2008).

A detailed investigation of the properties of the electronic excitation rates and the influence of the nonthermal character and properties of the nonthermal Lorentzian distribution function can also be found in the figures and tables. Figures 1 and 2 show the electronic excitation rate coefficients $\bar R_{\rm ex} ( {\equiv} R_{\rm ex} /R_0)$ for the electronic excitations X ¹Σ⁺_g → b ³Σ_u⁺, X ¹Σ⁺_g → a ³Σ_g⁺, and X ¹Σ⁺_g → B ¹Σ_u⁺ of molecular hydrogen by electron impact in low-temperature astrophysical Lorentzian plasmas as a function of the spectral index κ when $\bar {T} = 0.08$ and 0.1, respectively. From these figures, the nonthermal effect of the Lorentzian plasma suppresses the electronic excitation rates of molecular hydrogen in low-temperature regions such as $\bar {T} < 0.1$. Hence, we have found that the electronic excitation rate coefficients $\bar R_{b\,{}^3\Sigma _u^ + }$, $\bar R_{a\,{}^3\Sigma _g^ + }$, and $\bar R_{B\,{}^1\Sigma _u^ + }$ of H₂ in non-Maxwellian plasmas are smaller than those in Maxwellian plasmas in low-temperature regions. In addition, we have found that $\bar R_{b\,{}^3\Sigma _u^ + } > \bar R_{B\,{}^1\Sigma _u^ + } > \bar R_{a\,{}^3\Sigma _g^ + }$ in low-temperature Lorentzian astrophysical plasmas. It is interesting to find out that the X ¹Σ⁺_g → B ¹Σ_u⁺ electronic excitation rate coefficient is almost identical to the X ¹Σ⁺_g → a ³Σ_g⁺ electronic excitation rate coefficient for κ > 6. Figures 3–5 represent the excitation rate coefficients for the electronic excitations X ¹Σ⁺_g → b ³Σ_u⁺, X ¹Σ⁺_g → a ³Σ_g⁺, and X ¹Σ⁺_g → B ¹Σ_u⁺ of molecular hydrogen in intermediate-temperature regions of astrophysical Lorentzian plasmas as a function of the spectral index κ when $\bar {T} = 0.5$, 1, and 2, respectively. As shown in these figures, the nonthermal effect of the Lorentzian velocity distribution enhances the electronic excitation rates of the molecular hydrogen in intermediate-temperature regions such as $0.5 < \bar {T} < 2$. It is also found that the electronic excitation rate coefficients $\bar R_{b\,{}^3\Sigma _u^ + }$, $\bar R_{a\,{}^3\Sigma _g^ + }$, and $\bar R_{B\,{}^1\Sigma _u^ + }$ of H₂ in non-Maxwellian plasmas are greater than those in Maxwellian plasmas in intermediate-temperature regions. In addition, it is found that $\bar R_{b\,{}^3\Sigma _u^ + } > \bar R_{B\,{}^1\Sigma _u^ + } > \bar R_{a\,{}^3\Sigma _g^ + }$ in intermediate-temperature Lorentzian astrophysical plasmas. Hence, we have found that the X ¹Σ⁺_g → b ³Σ_u⁺ electronic excitation provides the largest contribution in low-temperature regions. Figures 6 and 7 represent the excitation rates for the electronic excitations X ¹Σ⁺_g → b ³Σ_u⁺, X ¹Σ⁺_g → a ³Σ_g⁺, and X ¹Σ⁺_g → B ¹Σ_u⁺ of the molecular hydrogen in high-temperature regions of astrophysical Lorentzian plasmas as a function of the spectral index κ when $\bar {T} = 5$ and 10, respectively. From these figures, it is interesting to see that the nonthermal property of the Lorentzian plasma enhances the X ¹Σ⁺_g → B ¹Σ_u⁺ excitation rate coefficient but suppresses the X ¹Σ⁺_g → b ³Σ_u⁺ and X ¹Σ⁺_g → a ³Σ_g⁺ excitation rate coefficients in high-temperature regions such as $\bar {T} > 5$. Hence, we can expect that the electronic excitation rate $\bar R_{B\,{}^1\Sigma _u^ + }$ of H₂ in non-Maxwellian plasmas are greater than those in Maxwellian plasmas in intermediate-temperature regions and the electronic excitation rate coefficients $\bar R_{b\,{}^3\Sigma _u^ + }$ and $\bar R_{a\,{}^3\Sigma _g^ + }$ of H₂ in non-Maxwellian plasmas are smaller than those in Maxwellian plasmas. In addition, it is shown that $\bar R_{B\,{}^1\Sigma _u^ + } > \bar R_{b\,{}^3\Sigma _u^ + } > \bar R_{a\,{}^3\Sigma _g^ + }$ in high-temperature Lorentzian astrophysical plasmas. As a result, the X ¹Σ⁺_g  →  B ¹Σ_u⁺ electronic excitation is found to be dominant in high-temperature regions. From Figures 3–7, it is interesting to see that $\bar R_{b\,{}^3\Sigma _u^ + } > \bar R_{B\,{}^1\Sigma _u^ + } > \bar R_{a\,{}^3\Sigma _g^ + }$ in intermediate-temperature Lorentzian astrophysical plasmas, however $\bar R_{B\,{}^1\Sigma _u^ + } > \bar R_{b\,{}^3\Sigma _u^ + } > \bar R_{a\,{}^3\Sigma _g^ + }$ in high-temperature Lorentzian astrophysical plasmas. In intermediate-temperature domains, the ordering of the rate coefficients $\bar R_{b\,{}^3\Sigma _u^ + } > \bar R_{B\,{}^1\Sigma _u^ + } > \bar R_{a\,{}^3\Sigma _g^ + }$ would be mainly determined by the magnitude of the excitation cross section and the electron distribution below the peak of the distribution function. However, in high-temperature domains, the excitation rate coefficients would be mostly determined by the magnitudes of the excitation cross sections and the high-energy tail of the electron distribution so that the rate coefficients in high-temperature Lorentzian astrophysical plasmas would be quite sensitive to the spectral index κ, i.e., the nonthermal character

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of the environment, because the electron distribution above the peak of the distribution function has a strong dependence on the spectral index of the Lorentzian plasma. Figure 8 represents the surface plot of the X ¹Σ⁺_g → b ³Σ_u⁺ electronic excitation rate coefficient $\bar R_{b\,{}^3\Sigma _u^ + }$ as a function of the electron temperature $\bar {T}$ and spectral index κ. As can be seen, the X ¹Σ⁺_g → b ³Σ_u⁺ electronic excitation rate coefficient shows the maximum peak near $\bar {T} = 2$. It is also found that the nonthermal character of the Lorentzian plasma enhances the peak of the X ¹Σ⁺_g → b ³Σ_u⁺ electronic excitation rate coefficient near $\bar {T} = 2$; however, it suppresses the peak of the X ¹Σ⁺_g → b ³Σ_u⁺ electronic excitation rate coefficient for $\bar {T} > 5$. In Tables 2–4, the individual X ¹Σ⁺_g → b ³Σ_u⁺ electronic excitation rate coefficients $\bar R_{b\,{}^3\Sigma _u^ + }$ of the molecular hydrogen per electron are given for various electron temperatures and spectral indices in astrophysical Lorentzian plasmas. Figure 9 shows the surface plot of the X ¹Σ⁺_g → a ³Σ_g⁺ electronic excitation rate coefficient $\bar R_{a\,{}^3\Sigma _g^ + }$ as a function of the electron temperature $\bar {T}$ and spectral index κ. As shown, the X ¹Σ⁺_g → a ³Σ_g⁺ electronic excitation rate shows the maximum peak near $\bar {T} = 2$ such as $\bar R_{b\,{}^3\Sigma _u^ + }$. In addition, it is found that

Table 2. The Excitation Rate Coefficients ${\bar{R}_{b{}^3\Sigma _u^ + }}$ of H₂ in Astrophysical Lorentzian Plasmas

Spectral Index	${\bar{R}_{b{}^3\Sigma _u^ + }}$ X1Σ+g → b3Σu+
(κ)	$\bar{T} = 0.08$	$\bar{T} = 0.1$	$\bar{T} = 0.5$	$\bar{T} = 1$	$\bar{T} = 2$	$\bar{T} = 5$	$\bar{T} = 10$
2	0.007014	0.009560	0.068040	0.120842	0.165702	0.164272	0.126117
3	0.005570	0.008827	0.108078	0.173099	0.189660	0.140317	0.094340
4	0.003789	0.006811	0.124644	0.190372	0.192002	0.130898	0.085713
5	0.002680	0.005363	0.133936	0.199036	0.192230	0.126147	0.081732
6	0.001992	0.004370	0.139920	0.204251	0.192076	0.123300	0.079443
7	0.001546	0.003672	0.144106	0.207736	0.191855	0.121407	0.077957
8	0.001244	0.003166	0.147201	0.210230	0.191639	0.120057	0.076915
9	0.001030	0.002787	0.149585	0.212104	0.191446	0.119047	0.076143
10	0.000873	0.002495	0.151477	0.213563	0.191277	0.118262	0.075549
Maxwellian	0.000080	0.000549	0.168015	0.225303	0.189355	0.112034	0.070979

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Table 3. The Excitation Rate Coefficients ${\bar{R}_{a{}^3\Sigma _g^ + }}$ of H₂ in Astrophysical Lorentzian Plasmas

Spectral Index	${\bar{R}_{a{}^3\Sigma _g^ + }}$ X1Σ+g → a3Σg+
(κ)	$\bar{T} = 0.08$	$\bar{T} = 0.1$	$\bar{T} = 0.5$	$\bar{T} = 1$	$\bar{T} = 2$	$\bar{T} = 5$	$\bar{T} = 10$
2	0.001144	0.001564	0.011583	0.021212	0.029945	0.029752	0.021641
3	0.000841	0.001346	0.018571	0.031370	0.035130	0.024581	0.014687
4	0.000530	0.000972	0.021509	0.034978	0.035767	0.022465	0.012789
5	0.000350	0.000720	0.023168	0.036852	0.035888	0.021384	0.011914
6	0.000243	0.000556	0.024240	0.038004	0.035899	0.020732	0.011410
7	0.000178	0.000446	0.024992	0.038785	0.035879	0.020296	0.011084
8	0.000136	0.000369	0.025548	0.039350	0.035852	0.019985	0.010855
9	0.000107	0.000313	0.025977	0.039777	0.035825	0.019751	0.010685
10	0.000087	0.000271	0.026317	0.040112	0.035800	0.019570	0.010555
Maxwellian	0.000003	0.000033	0.029297	0.042878	0.035467	0.018119	0.009551

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Table 4. The Excitation Rate Coefficients ${\bar{R}_{{B}{}^1\Sigma _u^ + }}$ of H₂ in Astrophysical Lorentzian Plasmas

Spectral Index	${\bar{R}_{{B}{}^1\Sigma _u^ + }}$ X1Σ+g → B1Σu+
(κ)	$\bar{T} = 0.08$	$\bar{T} = 0.1$	$\bar{T} = 0.5$	$\bar{T} = 1$	$\bar{T} = 2$	$\bar{T} = 5$	$\bar{T} = 10$
2	0.003622	0.004997	0.044252	0.097937	0.187308	0.339242	0.439726
3	0.001627	0.002674	0.056758	0.140542	0.264204	0.427342	0.504814
4	0.000765	0.001460	0.058662	0.154192	0.288814	0.452030	0.521117
5	0.000415	0.000900	0.058987	0.160867	0.300947	0.463731	0.528645
6	0.000252	0.000610	0.058944	0.164812	0.308171	0.470566	0.532996
7	0.000166	0.000444	0.058805	0.167414	0.312962	0.475048	0.535834
8	0.000117	0.000340	0.058649	0.169259	0.316372	0.478214	0.537832
9	0.000087	0.000272	0.058499	0.170633	0.318922	0.480570	0.539315
10	0.000067	0.000224	0.058362	0.171697	0.320902	0.482391	0.540460
Maxwellian	0.000001	0.000017	0.056577	0.180029	0.336606	0.496611	0.549364

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the nonthermal property of the Lorentzian plasma distribution enhances the X ¹Σ⁺_g → a ³Σ_g⁺ electronic excitation rate peak near $\bar {T} = 2$; however, it suppresses the peak of theX ¹Σ⁺_g → a ³Σ_g⁺ electronic excitation rate coefficient for $\bar {T} > 5$. In Tables 2–4, the individual X ¹Σ⁺_g → a ³Σ_g⁺ electronic excitation rate coefficients $\bar R_{a\,{}^3\Sigma _g^ + }$ of molecular hydrogen per electron are also given for various electron temperatures and spectral indices in astrophysical Lorentzian plasmas. Figure 10 shows the surface plot of the X ¹Σ⁺_g → B ¹Σ_u⁺ electronic excitation rate coefficient $\bar R_{B\,{}^1\Sigma _u^ + }$ as a function of the electron temperature $\bar {T}$ and spectral index κ. From this figure, it is found that the X ¹Σ⁺_g → B¹Σ_u⁺ electronic excitation rate coefficient shows no maximum peak and monotonically increases with an increase of $\bar {T}$ unlike the cases of $\bar R_{b\,{}^3\Sigma _u^ + }$ and $\bar R_{a\,{}^3\Sigma _g^ + }$. It is also found that the nonthermal character of the Lorentzian plasma enhances the X ¹Σ⁺_g → B ¹Σ_u⁺ electronic excitation rate coefficient near the peak for $\bar {T} > 1$. In Tables 2–4, the individual X ¹Σ⁺_g → a ³Σ_g⁺ electronic excitation rate coefficients $\bar R_{a\,{}^3\Sigma _g^ + }$ of molecular hydrogen per electron are also given for various electron temperature domains and spectral indices in astrophysical Lorentzian plasmas. Figure 11 shows the $\sigma _{B\,{}^1\Sigma _u^ + }$ cross sections versus relative velocity with a comparison to electron distributions for a low-kappa, high-kappa, and Maxwellian for a given temperature. As can be seen, the Maxwellian case is smaller than the nonthermal Lorentzian case near the peak of the cross section.

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In this paper, we investigated the X ¹Σ⁺_g → b ³Σ_u⁺, X ¹Σ⁺_g → a ³Σ_g⁺, and X ¹Σ⁺_g → B ¹Σ_u⁺ electronic transition processes of the molecular hydrogen in astrophysical Lorentzian plasmas. We obtained useful fitting formulae of the X ¹Σ⁺_g → b ³Σ_u⁺, X ¹Σ⁺_g → a ³Σ_g⁺, and X ¹Σ⁺_g → B ¹Σ_u⁺ electronic transition cross sections by electron impacts. In addition, we obtained the integral forms of the excitation rate coefficients $\bar R_{b\,{}^3\Sigma _u^ + }$, $\bar R_{a\,{}^3\Sigma _g^ + }$, and $\bar R_{B\,{}^1\Sigma _u^ + }$ for the X ¹Σ⁺_g → b ³Σ_u⁺, X ¹Σ⁺_g → a ³Σ_g⁺, and X ¹Σ⁺_g → B ¹Σ_u⁺ electronic transitions of the molecular hydrogen in astrophysical Lorentzian plasmas as functions of the spectral index and temperature of the Lorentzian plasma. It is found that the nonthermal character of the astrophysical Lorentzian plasma decreases the X ¹Σ⁺_g → b ³Σ_u⁺, X ¹Σ⁺_g → a ³Σ_g⁺, and X ¹Σ⁺_g → B ¹Σ_u⁺ electronic excitation rate coefficients of the molecular hydrogen in low-temperature regions: $\bar {T}\, {<}\, 0.1$. It is also shown that the nonthermal effect of the Lorentzian velocity distribution increases the X ¹Σ⁺_g → b ³Σ_u⁺, X ¹Σ⁺_g → a ³Σ_g⁺, and X ¹Σ⁺_g → B ¹Σ_u⁺ electronic excitation rate coefficients of molecular hydrogen in intermediate-temperature regions: $0.5 < \bar {T} < 2$. Moreover, it is perceived that $\bar R_{b\,{}^3\Sigma _u^ + } > \bar R_{B\,{}^1\Sigma _u^ + } > \bar R_{a\,{}^3\Sigma _g^ + }$ in low- and intermediate-temperature domains of the Lorentzian astrophysical plasma. It is interesting to see that the nonthermal character of the Lorentzian plasma increases the X ¹Σ⁺_g → B ¹Σ_u⁺ excitation rate coefficient but reduces the X ¹Σ⁺_g → b ³Σ_u⁺ and X ¹Σ⁺_g → a ³Σ_g⁺ excitation rate coefficients in high-temperature regions: $\bar {T} > 5$. In addition, it has been clear that $\bar R_{B\,{}^1\Sigma _u^ + } > \bar R_{b\,{}^3\Sigma _u^ + } > \bar R_{a\,{}^3\Sigma _g^ + }$ in high-temperature Lorentzian astrophysical plasmas. Hence, we have found that the nonthermal property of the plasma plays a crucial role in the electronic excitation processes of H₂ in astrophysical Lorentzian plasmas. These results provide useful information on the influence of the nonthermal character and properties of the astrophysical plasma on the X ¹Σ⁺_g → b ³Σ_u⁺, X ¹Σ⁺_g → a ³Σ_g⁺, and X ¹Σ⁺_g → B ¹Σ_u⁺ electronic excitation rate coefficients in nonthermal astrophysical Lorentzian plasmas. These results can be useful for understanding the electronic excitation processes of molecules in diffuse molecular clouds and for studying the structure and components of H₂-forming regions.

The authors gratefully acknowledge Dr. J.-S. Yoon and Dr. M.-Y. Song for enlightening discussions and useful data. Y.-D.J. also gratefully acknowledges Professor W. Roberge for useful discussions and warm hospitality while visiting the Department of Physics, Applied Physics, and Astronomy at Rensselaer Polytechnic Institute. This research was initiated while Y.-D.J. was affiliated with RPI as a visiting professor. This paper is dedicated to the late Professor R. J. Gould in memory of exciting and stimulating collaborations on atomic and molecular processes in astrophysical plasmas.

This research was supported by a Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (grant No. 2012–001493).