Kinetic analysis of the plasma sheath around an electron-emitting object with elliptic cross section

The structure of the sheath and the current exchange of two-dimensional electron-emitting objects with elliptic cross section immersed at rest in Maxwellian plasmas are investigated with an energy-conserving stationary Vlasov-Poisson solver free of statistical noise. The parameter domains for current collection within the orbital-motion-limited (OML) regime and current emission in space-charge-limited (SCL) conditions were studied by varying the characteristic dimension of the ellipse, its eccentricity, and the emission level. The analysis reveals the correlations between the onset of the non-OML and SCL regimes and the local curvature of the ellipse. As compared to nonemitting ellipses, electron emission broadens the parameter domain for OML current collection for ions and reduces considerably the current drop for non-OML conditions. Under identical plasma environments, elliptic bodies are more prone to operate under non-OML and SCL conditions than cylinders. Their emitted current in SCL conditions can be computed accurately from well-known results for cylinders if appropriate dimensionless variables and an equivalent radius are used. The role of the eccentricity, which acts as an integrability-breaking parameter, on the filamentation of the distribution function of the attracted species is studied.

Figure 1

Space charge at the $y=0$ [panel (a)] and $x=0$ [panel (b)] axes for several eccentricities.

Figure 2

Detail of the density of attracted ions inside the sheath for $e_p=0.75, {\delta }_i=1, {\varphi }_p=-5$.

Figure 3

Normal component of the electric field for $e_p=0.75, {\delta }_i=1, {\delta }_p=-5$.

Figure 4

Electrostatic potential at the $y=0$ [panel (a)] and $x=0$ [panel (b)] axes for several eccentricities. Collected current-to-OML current ratio vs the eccentricity [panel (c)].

Figure 5

Ratio $j_s\left(\theta \right)/j_s^{\mathrm{OML}}$ (a), curvature $k$ (b), and $dk/d\theta$ (c) vs the azimuthal angle $\theta$ for several eccentricities.

Figure 6

Ion distribution functions for $e_p=0.99$ (a) and 0.95 (c) evaluated at $\mathrm{\Gamma }$ and the value of $\theta$ where a minimum is reached for panel (a) in Fig. 4. Panel (b) shows two orbits for the initial conditions labeled with $A$ and $B$ in panel (a). Panel (d) displays the ion distribution function at $\mathrm{\Gamma }$ for a cylinder ($e_p=0$) of perimeter equal to the ellipse of panel (a).

Figure 7

Collected-to-OML current vs $b_p$ for several eccentricities. Panels (a) and (b) correspond to $\beta ={\delta }_{em}=0$ and $\beta =1.72$ and ${\delta }_{em}=0.32$, respectively.

Figure 8

Panel (a): normalized electrostatic potential profile vs ${(b_p/r)}^2$ for several emission levels at $x=0$ (dashed) and $y=0$ (solid). Panel (b): fraction of the perimeter under SCL conditions vs emission level $\beta$.

Figure 9

Normal component of the electric field for $e_p=0.75, {\delta }_i=1, {\varphi }_p=-4, {\delta }_{em}=0.32$.

Figure 10

Normalized distribution functions for emitted electrons ($f_{em}$) at $(x,y)=(0,b_p)$ [panels (a), (b) and (c)] and $(x,y)=(a_p=0)$ [panels (d), (e), and (f)] for $\beta =6$ [panels (a) and (d)], $\beta =7$ [panels (b) and (e)], and $\beta =10$ [panels (c) and (f)], respectively.

Figure 11

Emitted-to-half-Maxwellian current vs $\beta$.

Figure 12

Domains in the ${\epsilon }_i-\zeta$ plane of the ion distribution function at $(x,y)\approx (3.4,0)$ for $e_p=0$ (a), $e_p=0.75$ (b), $e_p=0.9$ (c), and $e_p=0.95$ (d). Panel (e) shows a detail of the region marked with a rectangle in panel (d). Orange, yellow, and blue colors identify $\mathrm{\Gamma }$-originated, ${\mathrm{\Gamma }}_{max}$-originated, and trapped orbits, respectively.