Editing Magnetic sail (section)

==== Coil attack angle effect on thrust and steering angle ====
[[File:Coil magnetic field orientation and forces.jpg|thumb|Coil magnetic field orientation and forces |upright=1.5]]
In 2005 Nishida and others published results from numerical analysis of an MHD model for interaction of the solar wind with a magnetic field of current flowing in a coil that momentum is indeed transferred to the magnetic field produced by field source and hence to the spacecraft.<ref name=":21">{{Cite journal |last1=Nishida |first1=Hiroyuki |last2=Ogawa |first2=Hiroyuki |last3=Funaki |first3=Ikkoh |last4=Fujita |first4=Kazuhisa |last5=Yamakawa |first5=Hiroshi |last6=Inatani |first6=Yoshifumi |date=2005-07-10 |title=Verification of Momentum Transfer Process on Magnetic Sail Using MHD Model |url=http://arc.aiaa.org/doi/abs/10.2514/6.2005-4463 |journal=41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit |language=en |location=Tucson, Arizona |publisher=American Institute of Aeronautics and Astronautics |doi=10.2514/6.2005-4463 |isbn=978-1-62410-063-5}}</ref> Thrust force derives from the momentum change of the solar wind, pressure by the solar wind on the magnetopause from equation {{EquationNote|MHD.1}} and Lorentz force from currents induced in the magnetosphere interacting with the field source. The results quantified the coefficient of drag, steering (i.e., thrust direction) angle with the solar wind, and torque generated as a function of attack angle (i.e., orientation) The figure illustrates how the attack (or coil tilt) angle <math>\alpha _t</math> orientation of the coil creates a steering angle for the thrust vector and also torque imparted to the coil. Also shown is the vector for the [[interplanetary magnetic field]] (IMF), which at 1 AU varies with waves and other disturbances in the solar wind, known as [[space weather]], and can significantly increase or decrease the thrust of a magnetic sail.<ref>[https://www.spaceweatherlive.com/en/help/the-interplanetary-magnetic-field-imf The Interplanetary Magnetic Field (IMF)], Space Weather Live.  Retrieved 11 February 2020.</ref>

For a coil with radial orientation (like a Frisbee) the attack angle <math>\alpha _t</math>= 0° and with axial orientation (like a parachute) <math>\alpha _t</math>=90°. The Nishida 2005 results<ref name=":21" /> reported a coefficient of drag <math>C_d</math> that increased non-linearly with attack angle from a minimum of 3.6 at <math>\alpha _t</math>=0 to a maximum of 5 at <math>\alpha _t</math>=90°. The steering angle of the thrust vector is substantially less than the attack angle deviation from 45° due to the interaction of the magnetic field with the solar wind. Torque increases from <math>\alpha _t</math>= 0° from zero at to a maximum at <math>\alpha _t</math>=45° and then decreases to zero at <math>\alpha _t</math>=90°. A number of magnetic sail design and other papers cite these results. In 2012 Kajimura reported simulation results<ref name=":31">{{Cite journal |last1=Kajimura |first1=Yoshihiro |last2=Funaki |first2=Ikkoh |last3=Matsumoto |first3=Masaharu |last4=Shinohara |first4=Iku |last5=Usui |first5=Hideyuki |last6=Yamakawa |first6=Hiroshi |date=2012-05-01 |title=Thrust and Attitude Evaluation of Magnetic Sail by Three-Dimensional Hybrid Particle-in-Cell Code |url=https://arc.aiaa.org/doi/10.2514/1.B34334 |journal=Journal of Propulsion and Power |volume=28 |issue=3 |pages=652–663 |doi=10.2514/1.B34334}}</ref> that covered two cases where MHD applicability occurs with <math>r_g/L</math>=1.125 and where a kinematic model is applicable <math>r_g/L</math>=0.125 to compute a coefficient of drag <math>C_d</math> and steering angle. As shown in Figure 4 of that paper when MHD applicability occurs the results are similar in form to Nishida 2005<ref name=":21" /> where the largest <math>C_d</math> occurs with the coil in an axial orientation. However, when the kinematic model applies, the largest  <math>C_d</math> occurs with the coil in a radial orientation. The steering angle is positive when MHD is applicable and negative when a kinematic model applies. The 2012 Nishida and Funaki published simulation results<ref name=":110">{{Cite journal |last1=Nishida |first1=Hiroyuki |last2=Funaki |first2=Ikkoh |date=May 2012 |title=Analysis of Thrust Characteristics of a Magnetic Sail in a Magnetized Solar Wind |url=https://arc.aiaa.org/doi/10.2514/1.B34260 |journal=Journal of Propulsion and Power |language=en |volume=28 |issue=3 |pages=636–641 |doi=10.2514/1.B34260 |issn=0748-4658}}</ref> for a coefficient of drag <math>C_D</math>, coefficient of lift <math>C_L</math> and a coefficient of moment <math>C_M</math> for a coil radius of <math>R_c</math>=100&nbsp;km and magnetopause radius <math>R_{mp}</math>=500&nbsp;km at 1 AU.