This section describes the PIC simulation configurations of a positively charged tether in the solar wind environment. The simulation code HiPIC, which was developed by the Japan Aerospace Exploration Agency's Engineering Digital Innovation Center (JEDI) , was used to perform this simulation. HiPIC is an electrostatic code that models 3-D rectangular cells in space and uses the full PIC method to calculate collisionless kinetic plasma. HiPIC solves Newton's equations of motion for each particle using the Buneman–Boris method and solves Poisson's equations to obtain the electric potential structure in the computational domain using a discrete sine transformation. HiPIC can be used to calculate the interaction between plasmas and the spacecraft, which is modeled with rectangular internal boundaries. A detailed description of the performance of the code is given in .

The simulation and physical parameters are given in Table . The electron density ne and the proton density np are ne=np=n0=1.0×107 m-3, and their temperatures are kBTe=100 eV and kBTp= 12 eV, respectively. The background plasmas have a solar wind drift velocity of vd=400 km s-1 along the x axis. All of the edges of the simulation domain were fixed to V=0V (Dirichlet boundary condition). The background plasma particles were injected from all the domain boundaries in each time step as many times as the number of outgoing particles in previous time step.

A tether-like rectangular model was set in the center of the computational domain, as shown in Fig. . The size of the model is 1×200×1 cells, which is equivalent to a tether with dimensions of rw= 15 cm and L=60 m. Tethers with L=30 and 75 m were also simulated to confirm the influence of the length of the tether on the thrust per unit length.

Because of the limitation of the calculation resources, we cannot include the emission of the electron beam from the tether's edge and simulate the self-charging of the tether. Instead of emitting the electron, the surface potential V0 of the tether was fixed to an inputted value. showed that effects of emitted electrons on the potential structure were small, so we consider that the absence of emitted electrons do not cause significant differences in the force acting on the tether.

V0 was varied from 0 to 4.0 kV, and the thrust acting on the tether Fx was calculated. Fx is the x component of the thrust and was calculated as the sum of the total momentum of the particles impinging on the tether during each time step divided by Δt and the Coulomb force calculated from the Maxwell stress tensor. The simulation progressed with time steps of Δt= 20 ns until the time variation of the external force became zero. In almost all the cases, the total iteration was 10 000 steps (= 0.2 ms). An additional 2000 steps were calculated for the V0=4.0 kV case.

To perform the computation, we applied an MPI parallelization and an OpenMP parallelization. Each case used 2048 cores on Cray XE4 for calculation (1024 processes for MPI, two threads for OpenMP), requiring approximately 12 h to run.

Figure a shows the time variation of Fx at V0=1.3 kV. At the beginning of the simulation, Fx was almost zero because the ambient particles had not yet begun to respond to the electrostatic potential of the tether. Fx then increased with time and converged to a specific value. At the end of the simulation, Fx was 0.35, 0.76, and 0.96 µN for L=30, 60, and 75 m, respectively. These values represent the total force acting on the tether, including the sum of the particles hitting the whole tether and the force calculated from the Maxwell stress tensor. However, the thrust per unit length indicates the performance of the E-sail; thus, Fx/L was calculated, as shown in Fig. b. Fx/L was 11.7, 12.7, and 12.8 nN m-1 for L=30, 60, and 75 m, respectively. The thrust per unit length for L=30 m was slightly smaller than that for 60 and 75 m. This lower value of Fx/L may be due to the end effect or the effect which arises when L is small in comparison with λD. At L=60 and 75 m, the values of Fx/L were almost equal; thus, L=60 m is considered to be sufficiently long to simulate an infinite tether.

The kinks in the thrust between t=10 and 20 µs shown in Fig.  correspond to collections of ambient electrons. Figure  shows the time history of the current on the surface of the tether with L=60 m. The ambient electron current (purple line) varied dramatically between t=10 and 20 µs. This is an initial response of the ambient electrons to the potential of the tether. The electron plasma frequency ωp was approximately 178 kHz so the response time of ambient electrons were approximately 5.6 µs. Collected electrons do not directly contribute to the thrust, because their momentum is small. Instead, they temporarily shield the potential structure more strongly than in a steady state, causing the thrust to stop increasing with time, as shown in Fig. .

Figure  also reveals that the dominant source of the force acting on the tether was the force from the Maxwell stress tensor, not from the protons hitting the tether because the proton current was almost zero. This fact shows that the computation successfully simulates the thrust generation by proton deflection.

To compare the present results with previous thrust estimations, Fx/L was calculated at various V0 values, as shown in Fig. . The blue line in Fig.  is the result of our PIC simulation with L=60 m and rw=15 cm. The black and red dotted lines show the thrust estimated by and , respectively, with rw=15 cm. In the present simulation, Fx/L was 67.1 nN m-1 at V0=4.0 kV; in contrast, and estimated Fx/L to be 123 and 246 nN m-1, respectively. These results indicate that the thrust characteristics of the E-sail are different from those of conventional estimations. This difference is likely due to the difference in the potential structure around the tether.

Figure  compares the electrostatic potential structures obtained in the present study and two previous studies at V0=1.0, 2.0, 3.0, and 4.0 kV. The purple lines show the PIC simulation results for the potential at the center of the tether (y=76.8 m, z=76.65 m). At all potentials, the potential obtained from the present PIC simulations was lower than the potentials obtained using Eqs. () and (). Figure  indicates that potential shielding by ambient electrons is not appropriately included in Eqs. () and (). Figure  also shows that the sheath length assumed by Eq. () is consistent with that obtained by PIC simulation.

Figure  shows the proton density structure at t=0.2 ms. At V0=2.0, 3.0, and 4.0 kV, a zero-density (np=0 m-3) region was present in front of the tether. No protons impinged on the surface of the tether. The momentum of the protons was transferred to the tether through the Coulomb force acting on the tether. In contrast, there was no zero-density region in front of the tether at V0=1.0 kV. This is because the radius rw of the tether is relatively large. The drift velocity of the proton (400 km s-1) is equivalent to 0.83keV, but the figure indicates that V0=1.0 kV is not sufficient to deflect all of the protons for rw=15 cm. Although a wake region is present at V0=1.0 kV, it was formed by protons impinging on the surface of the tether. At a very small rw, a zero-density region may appear for V0=1.0 kV.

This study then considered the high-density proton region in front of the tether. Figure  shows the electron and proton density structure along the x axis at V0=2.0 kV. As with Fig. 3a in , Fig. b reveals a high-density region in front of the tether. The maximum proton density was 2.0×107 m-3 at V0=2.0 kV. The maximum electron density was approximately 1.0×108 m-3 at V0=2.0 kV. The maximum positive charge density of the high-density region is one fifth of the negative charge density of the electrons. The high-density region does not compensate for the potential; that is, it does not reduce the shielding effect of the electrons. Trapped electron removal, which was discussed by , was not observed in the present simulation. Thus, the thrust models given by Eqs. () and () are inappropriate for estimating the thrust of the E-sail, and a new model considering appropriate potential shielding must be developed.

The Poisson equation in plasma is expressed as ∇2V(r)=-ρ(r)ε0-eε0(np(r)-ne(r)), where ρ is the space charge density. From the velocity distribution function of electrons, we assume that the electron's density distribution becomes the Boltzmann distribution: ne(r)=n0exp⁡eV(r)kBTe. Assuming that the normalized potential eV(r)/kBTe becomes small in the distance and also assuming that np=n0, Eq. () becomes ∇2V(r)=-ρ(r)ε0+e2n0ε0kBTeV(r) with the first-order approximation. Defining kD=e2n0/ε0kBTe, we obtain (∇2-kD2)V(r)=-ρ(r)ε0.

The 1-D solution of Eq. () is well known and is called the Yukawa potential. However, the 2-D analytical solution of Eq. (), which would represent the shielded potential structure around an infinite tether, remains unknown. This is why previous studies had to assume an artificial potential structure, such as those given by Eqs. () and ().

developed a numerical method of calculating the potential structure around a tether in plasma, adding the term h2kD2 to the solution of the difference equation corresponding to the differential equation given by Eq. () as follows Vm,n=h2ε0ρm,n4+h2kD2-Wm-W-m-Wn-W-n,W=e2πi/N, where Vm,n is the electrostatic potential solution in k space at the grid point (m,n), N is the total number of cells in 2-D space, and h is the cell width used in the Fourier transformation. The 2-D inverse FFT of Eq. () was taken to obtain the shielded electrostatic potential in plasma, and the solution was found to be consistent with the potential given by the full PIC simulation.

The proposed estimation method was used to obtain a realistic potential without performing the full PIC simulation. To realize an equipotential within the radius of the tether, the capacity matrix method was also used . Figure  compares the potential structures obtained using the 2-D FFT method with the PIC results. A cell width of h=0.03 and N=8192 cells in 2-D space were used for the Fourier transformation. The potential structures estimated using the proposed method were consistent with those obtained from the PIC simulation results (Fig. ). At V0=1 kV, there was a small difference (approximately 20 V) between the potential structures estimated using the proposed method and the PIC simulation results behind the tether, but this difference did not cause a difference in the thrust, because only the potential in front of the tether contributes to the thrust.

Method 1 considers the effective radius rs that satisfies the following equation: eV(rs)=12mpvd2, where rs is the distance from the tether at which the electrostatic potential energy is equal to the kinetic energy of the drifting proton. assumed that the scattering cross section is proportional to rs with a coefficient of proportionality of K, meaning the thrust per unit length dF/dL can be expressed as dFdL=KrsPdyn=Krsn0mpvd2, where Pdyn=mpn0vd2 is the dynamic pressure of a solar wind proton.

In , V(r) is given by Eq. (). In the present study, V(r) was replaced with the numerical solution obtained using the 2-D FFT method, which considers potential shielding by ambient electrons. For the numerical calculation, the value of r that minimizes the difference between the potential energy and the kinetic energy f1, which is expressed as f1(r)=eV(r)-12mpvd2, is used as rs instead the value of r. The coefficient of proportionality was set to K=3.09, as obtained by .

The thrust modeling method by is based on Coulomb scattering. The following equation is formed by adding an angular momentum term to Eq. (): eV(r)+Lm22mpr2=12mpvd2, where Lm=mpvdρ is the angular momentum and ρ is the impact parameter. Defining Esw=12mpvd2 yields eV(r)Esw+ρ2r2=1. Additionally, l and Ueff were defined as l≡ρrUeff(l)≡eV(r)Esw+l2. Ueff(l) is equal to the left-hand side of Eq. (), meaning the distance rmin that minimizes f2(ρr)=1-Ueff(ρr) is equivalent to the distance rs obtained using the effective radius method. When the scattering angle is defined as χ(ρ)=π-δ(ρ), as shown in Fig. 4 of , the following equations give χ(ρ): δ(ρ)=ρ∫rminrmaxdrr21-ρ2r2-eV(r)Esw+∫rmax∞drr21-ρ2r2=δ1(ρ)+δ2(ρ)δ1(ρ)=∫lminlmaxdl1-Ueff(l)δ2(ρ)=∫0lmindl1-l2=sin⁡(lmin), where lmax=ρ/rmin and lmin=ρ/rmax. rmax denotes the maximum length that scattered particles can be affected by an electrostatic potential. rmax was defined as rshe-b by and has been calculated as follows by : 1.531-2.56λDrsh4/5rshλD4/3ln⁡rshrw=eV0kBTe.

Then, the thrust per unit length can be expressed as dFdL=n0vd×2∫0rmax2mpvdsin⁡χ(ρ)2cos⁡π-χ(ρ)2dρ=4Pdyn∫0rmaxsin⁡2χ(ρ)2dρ.

In the present estimation, V(r) in Eq. () was replaced with the numerical solution obtained using the 2-D FFT method, and the maximum affection length rmax=2.0λD was used instead of the value of rmax=rshe-b used by .

The thrust per unit length was obtained for V0=0 to 4.0 kV with rw=15 cm. Figure  shows the thrust estimated using the effective radius method (purple line) and the Coulomb scattering method (green line). These modified thrust estimations are similar to the thrust obtained from the PIC simulation, meaning they are also significantly lower than estimations from previous studies. The modified estimated thrusts obtained using Methods 1 and 2 are 79.1 and 62.0 nN m-1, respectively. The thrust estimated using Method 2 at 4.0 kV is in very good agreement with the PIC result of 67.1 nN m-1. It should be noted that the difference between the modified and previous estimation methods is simply the electrostatic potential structure. The effective proton deflection area with shielding is smaller than that without the shielding. Thus, the inclusion of potential shielding results in reduced thrust.

These two modified estimation methods yielded similar estimated values of the thrust per unit length. Thus, the two estimation procedures were shown to be essentially similar, and the difference between the original estimated thrust values was revealed to have been caused by differences in the potential structures. The effective radius method contains an approximation coefficient K, whereas the Coulomb scattering method does not have contain an approximation coefficient; thus, the Coulomb scattering method was considered to be more consistent with the PIC results.