University of Kansas
Several test runs of the model were carried out in order to determine the sensitivity of the results to some key physical processes. Ion-ion and electron-ion collisional coupling terms were omitted from the three energy equations in one test run, yet all calculated quantities were the same to within 1%. In another test run, ion production of the medium mass species was set equal to zero for radial distances greater than ~ 2.5 RT. Again, calculated quantities differed from the standard model results by less than a couple percent, indicating that mass-loading in an "extended" exosphere is not as important as processes (including mass-loading) occurring closer to Titan.
One difficulty in setting up the model was the choice of ion-neutral collision frequency for the momentum equation -- should the polarization interaction value be used or a charge-transfer value (about a factor of 4 greater) be used. The latter is more relevant outside the ionosphere than in the magnetosphere, whereas the former is more relevant in the ionosphere. Our standard model uses the larger charge-transfer value, but a test run in which a smaller value (i.e., a factor of 4 smaller) was used for the medium-medium momentum transfer collision frequency demonstrated that the choice is not too critical. The flow speed was only about 10% higher in the 4000 - 5000 km region when the lower value was used, and outside this region the difference was even less than this.
It appears that what is most important to the dynamics in the near-Titan region (r < 2 RT) is ion-neutral friction and, to a lesser extent, mass-loading. The resulting slow-down of the flow results in the formation of a magnetic barrier which is confined to the near-Titan region. This magnetic barrier acts as the real obstacle to the external flow. On the bottomside of the magnetic barrier, the thermal ionospheric pressure is important but is not quite sufficient to balance the peak magnetic pressure. The maximum thermal pressure is about half of the magnetic pressure. Ion-neutral collisional momentum transfer is especially important in the lower ionosphere. The momentum balance at Titan appears to have analogies with the solar wind interaction with either Mars or with Venus at times when the solar wind dynamic pressure is high [cf. Shinagawa and Cravens, 1988, 1989], in which case, the ionospheric density falls off gradually rather than at a "sharp" ionopause.
Magnetic field lines cannot be curved in a two-dimensional MHD model such as the one described in this paper, yet we know from the Voyager magnetometer observations [Ness et al., 1982] that downstream of Titan, in the wake, the field lines are strongly draped. Our claim that our 2D MHD model is basically valid in the ram and flank regions, where field line curvature is less important, is supported by a comparison of magnetic field strengths calculated from a two (Figure 7) and a three-dimensional MHD model. The 3D model [Ledvina and Cravens, 1998], like the 2D model, shows the development of a magnetic barrier, but unlike the 2D model it also shows strong field line draping in the wake region. The field strength patterns of the 2D and 3D models are quite similar in the ram and flank directions. The advantage of the 2D model is that, with its 3 ion species and relatively high spatial resolution, it can reproduce a reasonable ionosphere, which a single-fluid model cannot do.
The magnetic barrier can be considered to be the main obstacle to the external flow. Within the barrier, magnetic pressure plays a key role dynamically, but upstream the plasma beta is high and the flow is superAlfvenic suggesting that magnetic forces are not important. The upstream flow is also subsonic and submagnetosonic, so that no bow shock is expected. Indeed, a bow shock did not appear in the model results. Being subsonic, the flow is able to gradually adjust to the presence of the obstacle.
Mass-loading outside a distance of about 2 RT (in the fast flow region) also does not appear to be important relative to other dynamical processes. That is, the interaction of the external plasma with Titan is not primarily cometary in nature [cf. Galeev, 1986]. This was demonstrated by running the model with and without ion production for distances greater than 3 RT; the results were almost the same. One can also estimate the importance of mass-loading by evaluating the expression for the momentum flux from Galeev  (or from one of the many other solar wind-comet interaction papers) but using the production rates shown in Figure 2. For r = 2.5 RT, the momentum flux divided by the upstream value, that is ru / (rinfinityuinfinity), is equal to approximately 1.01, whereas significant flow perturbations such as a mass-loading induced shock requires this ratio to be about 1.1- 1.3. Note that rinfinity is the upstream mass density and uinfinity is the upstream flow speed. On the other hand, for distances within 2 RT or so, mass-loading due to ion production becomes important as does ion-neutral friction. Why is the outer boundary of the slow-flow region (we can call it the "critical radius") located where it is (r ~ 2 RT)? This location is where the neutral density is first high enough that the momentum balance is affected by collisional processes (i.e., mass-loading due to ionization of neutrals and ion-neutral friction). The critical neutral density (nnc) can be roughly estimated by equating the cumulative collisional momentum change following a flow streamline down the ram direction with the momentum flux of the flow just outside the slow-flow boundary (this is just the dynamic pressure rcuc2 = ncmuc2 where nc is the ion number density -- we will just work with the medium and heavy species due to their larger masses). The cumulative collisional momentum change is approximately mucL (Pc + 0.5 ncninc) where Rc and ninc are the ion production and ion-neutral collision frequency at the critical location and L ~ 1000 km is a neutral length scale at that location. The mass-loading term initially dominates and Rc ~ I nnc with an ionization frequency I ~ 10-8 s-1. The critical neutral density is then of the order of nnc ~ ncuc / (L I) ~ 1-3 x 105 cm-3, where nc ~ 1 cm-3 and uc ~ 5 km/s. This value of the (medium plus heavy) neutral density is found at r ~ 2 RT.
The flow pattern outside a radial distance of ~ 2.5 RT has a qualitative appearance to the flow pattern around a cylinder predicted by classical potential fluid theory [cf. Paterson, 1983]. This is not entirely surprising in that the flow is subsonic and approximately incompressible. Making the further assumption that the flow is irrotational, a two-dimensional stream function, Y, can be determined. Y satisfies Laplace's equation. For flow around a cylinder of radius, a, the stream function and flow velocity are given by:
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