for the magnetic moment into equation yields the following: When the magnetic field source strength is specified, substituting from the pressure balance analysis from equation into the above and solving for yields the following: This is the expression for when with from equation (4), with from equation (4), and the magnetopause distance of the Earth. This equation shows directly how a decreased falloff rate dramatically increases the effective sail area for a given field source magnetic moment and determined from the pressure balance equation . Substituting this into equation yields the plasma wind force as a function of falloff rate , plasma density (kg/m3), coil radius (m), coil current (A) and plasma wind velocity (m/s) as follows: using equation for and equation for . This is the same expression as equation (10b) when and , equation (108) and equation (5) with other numerical coefficients grouped into the term. Note that force increases as falloff rate decreases. When the design target is the magnetopause radius , the required field source strength is then determined directly from equation as follows:which then determines the magnetic moment from equation and plasma wind force from equation . General kinematic model When the MHD applicability test of <1 then a kinematic simulation model more accurately predicts force transferred from the plasma wind to the spacecraft. In this case the effective sail blocking area < . The left axis of the figure is for plots of magnetic sail force versus characteristic length . The solid black line plots the MHD model force from equation . The green line shows the value of ion gyroradius 72 km from equation . The dashed blue line plots the hybrid MHD/kinematic model from equation from Fujita04. The red dashed line plots a curve fit to simulation results from Ashida14. Although a good fit for these parameters, the curve fit range of this model does not cover some relevant examples. Additional simulation results from Hajiwara15 are shown for the MHD and kinematic model as single data points as indicated in the legend. These models are all in close agreement. The kinematic models predict less force than predicted by the MHD model. In other words, the fraction of thrust force predicted by the MHD model is lost when as plotted on the right axis. The solid blue and red lines show for Fujita04 and Ashida18 respectively, indicating that operation with less than 10%