Electron impact ionization

The magnetospheric electrons and ions impinging Callisto’s atmosphere are a second ma-jor source for the generation of ionospheric particles. The respective rate coefficients for the ion impact ionization are, however, significantly lower than the rates for the electron impact ionization. At Io and Europa electron impact ionization is even the most important mass loading process (Saur et al. 1998, 1999). The production rate for the electron impact ionization of the neutral species ns can be calculated from:

P_imp,ns = f_imp,ns(T_e)n_en_ns, (3.31)

where the ionization rate f_imp,nsfor a specific electron temperature of T_ecan be written as:

f_imp,ns(Te) =

denotes the electron velocity given at an energy E. The normalized distribution function for the electrons in energy space

Fe(E) = 2√ E

π (kBTe)⁻³²e^−kBTeE (3.34) is assumed to be Maxwellian for our purposes. σ_imp,ns are the electron impact ioniza-tion cross secioniza-tions at different energies. In this work we apply the so called BEB cross

section model put forward by Hwang and Kim (1996) to perform a numerical integra-tion of Equaintegra-tion (3.32). The impact ionizaintegra-tion rates for both neutral species used in our model are shown in Figure 3.7. For the magnetospheric electrons with a temperature of kBT_e,ms,0 =635 eV and an average density of n_e,ms,0 ≈ 0.1 cm⁻³the respective production rates at Callisto’s surface are P_imp,ms,CO2 = 6 cm⁻³ s⁻¹ and P_imp,ms,O2 = 73 cm⁻³s⁻¹. Due to the decreased surface neutral densities in our model these values are diminished by the factor 1/H_rel=1/10 for our simulations. For the ionospheric electrons with Te,is =300 K the rate drops almost to zero (several decades of magnitudes lower) even though their den-sities are significantly higher. If we compare these production rates with the rates given in Section 3.2.2.1 it turns out that the photo ionization process at Callisto dominates by a factor of about three. However, the impact ionization process generates additional ion-ized particles in the geometrical shadow region of Callisto. Therefore, electron impact ionization still provides an important contribution to the overall ionospheric distribution.

Above calculations crucially depend on the temperature of the electrons Te. Due to the nearly infinite heat conduction along the field lines the magnetospheric electrons which are cooled down during the interaction can be effectively reheated by the energy reservoir of the electrons in the other parts of the plasma torus. To account for this effect we introduce an effective ionization rate f_imp,max,nsby equating:

E_loss,imp = f_imp,max,nsn_e,0E_ion,effn_ns,04π

E_loss,imp denotes the total energy lost due to the impact ionization throughout the entire at-mosphere. The total neutral density is given by the integral in Equation (3.35). n_ns,0is the surface density of the neutral species. An effective ionization energy E_ion,eff = ³₂kBT_ion,eff is used instead of the neutral species’ ionization threshold potentials (E_ion,CO2 = 13.77 eV and E_ion,O2 = 12.06 eV) here. We do this to account for other potential energy loss pro-cesses occurring during electron impacts. Values of E_ion,eff,CO2 = 35 eV and E_ion,eff,O2 = 32 eV are given by Bauer (1973) and Banks and Kockarts (1973).

In the second Equation (3.36) E_torus denotes the energy reservoir within the torus which is available to heat the electrons involved in the plasma interaction. The electron energy influx is given in terms of the magnetospheric electron temperature T_e,0 = 635 eV which is assumed to be constant along the field lines. v₀is the velocity of the impinging plasma particles. The integral over the torus electron density along the field lines connected to Callisto is given for a torus scale height of Hcs = 3.65 RJ (see Section 3.1.2). The width

3.2 Model description

Figure 3.7:Electron impact ionization rates fimp,nsin10⁻⁷cm³s⁻¹vs. electron temperature in eV for O₂and CO₂.

of the respective flux tubes (2 RC) is diminished by the factor α_wing. As the electron flow follows the isolines of the electric potential, the fraction of the magnetospheric electrons impinging Callisto’s interaction region is defined by the last isoline connected to this region (see Figure 3.8). For a strong interaction or for high Pedersen conductances the width of the flux tubes decreases. This is reflected by the factor α_wing which we already introduced as α in Section 2.3.1.2. Apart from the definition of given in Equation (2.31) we can ”measure” the relative interaction strength by noting that α_wing = v_wing/v₀, where vwingdenotes the diminished plasma velocity inside the Alfvén wings. For our simulations we determine v_wing at a distance of 3 RC from Callisto’s center following both Alfvén characteristics (Section 1.2.2, Equation 1.3).

For our interaction model we use an approach which incorporates both methods presented above to determine the electron impact ionization rate. Using α_wingwe can determine the maximum possible ionization rate f_imp,max,ns assuming infinite heat conduction along the field lines. However, an upper limit for the applicable ionization frequency is defined by the ionization frequency of the background plasma f_imp,0,ns = f_imp,ns(T_e,ms,0) i.e.,

f_imp,ns=









f_imp,max,ns for f_imp,max,ns < f_imp,0,ns

f_imp,0,ns for f_imp,max,ns > f_imp,0,ns. (3.38) For the carbon dioxide atmosphere the energy within the torus allows for a maximum impact ionization rate of f_imp,max,CO2 = 4.87 × 10⁻⁶cm³ s⁻¹α_wing, but the upper limit for the ionization assuming a homogeneous electron plasma with kBT_e,ms,0 = 635 eV and n_e,ms,0 = 0.1 cm⁻³is f_imp,0,CO2 = 1.4 × 10⁻⁷cm³ s⁻¹. The respective values for molecular oxygen are f_imp,max,O2 = 3.08×10⁻⁷cm³s⁻¹α_wingand f_imp,0,O2 = 1.04×10⁻⁷cm³s⁻¹. At the start of our simulations v_wing= v₀implies that α_wing =1 and f_imp,0,nsis used to initialize the impact ionization term. It should be already noted at this stage that typical values for α_wing encountered within our simulations lie between 0.01 and 0.1 (see Section 3.3). Therefore, the impact ionization rate is set to f_imp,max,nsα_wingafter a certain simulation time.

Figure 3.8: Contours of the electric potential outside an interaction region (gray circle) with constant ionospheric conductances after Saur (2004). The ambient plasma flows in the positive x-direction and the magnetic field points in the negative z-direction. The diameter of the flux tubes reaching the interaction region is diminished up to a value of2Rαwing, with the radius of the satellite R. For the significance of the additionally denoted parameters the reader is referred to the original publication.