Estimated Mass

The dust grain charge $Q_d$ is computed from the capacitance of the dust particles,

$$Q_d = 4\pi \epsilon_0 a U.$$ (4)

Assume that the dust is composed of 1 micron diameter solid spherical silica particles with a mass density of 1500 kg/m$^3$. The mass of an individual particle will be $m_d \sim 7.85 \times 10^{-16}$ kg. The balance of the charging equations

$$I_e = I_I +I_\nu$$ (5)

under the constraint that the total charge in the plasma is conserved

$$en_i = en_e + Q_d n_d$$ (6)

allows an estimate of the average charge on the dust grains to be made. This calculation is made using plasma parameters for a hydrogen plasma generated by a spacecraft that is located at D = 1 AU, temperatures, $T = T_e = T_i \sim 1$ eV, and $T_d$ (dust temperature) $\sim 0.5$ eV, densities, $n_e = n_I \sim 10^7 m^{-3}$, and dust densities in the magnetic balloon of $n_d \sim 10^5 m^{-3}$. Under these assumptions, the dust grain charge will be negative, $Q_d \sim 5.4 \times 10^{-17} C$ or $Z_d = (Q_d/e) = 336$ electronic charges. The total mass of dust contained in a 20 km diameter magnetic balloon is $M_{\rm total} \sim 330$ kg. Finally, this gives a mass loading for the sail and a 500 kg payload (as computed from the ratio of total mass to projected circular surface area of a spherical magnetic balloon) of 0.0026 gm/m$^2$.

Figure 3 shows the total dust mass and dust grain charge as a function of the dust density. Note that decreasing the dust grain charge has the effect of increasing the gyro-orbit of the dust particles about the magnetic field, thereby reducing their confinement.