Editing Magnetic sail (section)

=== Magnetohydrodynamic model ===
Magnetic sail designs operating in a plasma wind share a theoretical foundation based upon a [[Magnetohydrodynamics|magnetohydrodynamic]] (MHD) model, sometimes called a fluid model, from [[Plasma (physics)|plasma physics]] for an artificially generated [[magnetosphere]]. Under certain conditions, the plasma wind and the magnetic sail are separated by a [[magnetopause]] that blocks the charged particles, which creates a drag force that transfers (at least some) momentum to the magnetic sail, which then applies thrust to the attached spacecraft as described in Andrews/Zubrin,<ref name=":4">{{Cite journal |last1=Andrews |first1=Dana |last2=Zubrin |first2=Robert |date=1990 |title=MAGNETIC SAILS AND INTERSTELLAR TRAVEL |url=http://path-2.narod.ru/design/base_e/msit.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://path-2.narod.ru/design/base_e/msit.pdf |archive-date=2022-10-09 |journal=Journal of the British Interplanetary Society |volume=43 |pages=265–272 |via=semanticscholar.org}}</ref> Cattell,<ref name="05ja026_full">{{cite journal |last1=Cattell |first1=C. |date=September 2005 |title=Physics and Technology of the Feasibility of Plasma Sails |url=https://www.researchgate.net/publication/249977236 |journal=Journal of Geophysical Research}}</ref> Funaki,<ref name=":3" /> and Toivanen.<ref name=":8" />

A plasma environment has [[Plasma parameters#Fundamental plasma parameters|fundamental parameters]], and if a cited reference uses cgs units these should be converted to SI units as defined in the NRL plasma formulary,<ref name=":33" /> which this article uses as a reference for plasma parameter units not defined in [[SI Units|SI units]]. The major parameters for plasma mass density are: the number of ions of type <math>i</math> per unit volume <math>n_i</math> the mass of each ion type accounting for isotopes <math>m_i</math> and the number of electrons  <math>n_e</math> per unit volume each with [[electron mass]] <math>m_e</math>.<ref name=":12">{{Cite journal |last=Wiesemann |first=K. |date=2014-04-02 |title=A Short Introduction to Plasma Physics |journal=CERN Yellow Report CERN-2013-007 |pages=85–122 |arxiv=1404.0509 |doi=10.5170/CERN-2013-007.85}}</ref> An average plasma mass density per unit volume for charged particles in a plasma environment <math>pe</math> (<math>sw</math> for stellar wind, <math>pi</math> for planetary ionosphere, <math>im</math> for interstellar medium) is expressed in equation form from [[Magnetohydrodynamics#Units|magnetohydrodynamics]] as<math>\textstyle \rho_{pe} = n_e m_e + \sum_{i}n_i m_i</math>. Note that this definition includes the mass of neutrons in an ion's nucleus. In SI Units per unit volume is [[Cubic metre|cubic metre (m<sup>−3</sup>)]], mass is [[kilogram]] (kg), and mass density is [[Kilogram per cubic metre|kilogram per cubic metre (kg/m<sup>3</sup>)]].
[[File:Artificial Magnetosphere Math Model.jpg|thumb|upright=1.5|Artificial Magnetosphere Model of Basic Magnetic Sail]]
The figure depicts the MHD model as described in Funaki<ref name=":3" /> and Djojodihardjo.<ref name=":0" /> Starting from the left a plasma wind in a plasma environment (e.g., stellar,  ISM or an ionosphere) of effective velocity  <math display="inline">v_{pe}</math> with density <math display="inline">\rho_{pe}</math> encounters a spacecraft with time-varying velocity <math display="inline">v_{sc}</math> that is positive if accelerating and negative if decelerating.  The apparent plasma wind velocity from the spacecraft's viewpoint is <math display="inline">u_{pe}=v_{pe}-v_{sc}</math>. The spacecraft and field source generate a [[magnetic field]] that creates a magnetospheric bubble extending out to a magnetopause preceded by a bow shock that deflects electrons and ions from the plasma wind. At the magnetopause the field source magnetic pressure equals the kinetic pressure of the plasma wind at a standoff shown at the bottom of the figure. The characteristic length <math display="inline">L</math> is that of a circular sail of effective blocking area <math>S=\pi \, R_{mp}^2</math>  where <math>R_{mp} \approx L </math> is the effective magnetopause radius. Under certain conditions the plasma wind pushing on the artificial magnetosphere bow shock and magnetopause creates a force <math display="inline">F_w</math> on the magnetic field source that is physically attached to the spacecraft so that at least part of the force <math display="inline">F_w</math> causes a force <math display="inline">F_{sc}</math> on the spacecraft, accelerating it when sailing downwind or decelerating when sailing into a headwind. Under certain conditions and in some designs, some of the plasma wind force may be lost as indicated by <math display="inline">F_{loss}</math> on the right side.

All magnetic sail designs assume a standoff between plasma wind pressure <math>p_w</math> and magnetic pressure <math display="inline">p_B</math> with SI units of [[Pascal (unit)|Pascal (Pa, or N/m<sup>2</sup>)]] differing only in a constant coefficient <math>C_{SO}</math> as follows:

{{NumBlk2|:|<math>p_w=\rho u^2 =p_B=\frac{B_{mp}^2}{C_{SO} \mu _0}</math>|MHD.1}}where <math display="inline">u</math> is the apparent wind velocity and <math>\rho</math>  is the plasma mass density for a specific [[Magnetic Sail#Modes of operation|plasma environment]],  <math>B_{mp}</math> the magnetic field flux density at [[magnetopause]], ''μ<sub>0</sub>'' is the [[vacuum permeability|vacuum permeability (N A<sup>−2</sup>)]] and <math display="inline">C_{SO}</math> is a constant that differs by reference as follows for <math display="inline">C_{SO}=2</math> corresponding to <math display="inline">p_w</math> modeled as [[dynamic pressure]] with no magnetic field compression,<ref name=":8" />  <math display="inline">C_{SO}=1</math> for <math display="inline">p_w</math> modeled as [[ram pressure]] with no magnetic field compression<ref name=":13" /><ref name="Slough2006" /> and <math display="inline">C_{SO}=1/2</math> for <math display="inline">p_w</math> modeled as ram pressure with magnetic field compression by a factor of 2<ref name=":3" />  Equation {{EquationNote|MHD.1}} can be solved to yield the required magnetic field <math>B_{mp}</math> that satisfies the pressure balance at magnetopause standoff as:

{{NumBlk2|:|<math display="block">B_{mp}=u \sqrt{\rho \, \mu_0 \, C_{SO} }</math>|MHD.2}}The force with SI Units of [[Newton (unit)|Newtons (N)]] derived by a magnetic sail for a plasma environment is determined from MHD equations as reported by principal researchers Funaki,<ref name=":3" /> Slough,<ref name="Slough2006" /> Andrews and Zubrin,<ref name=":4" /> and Toivanen<ref name=":8" /> as follows:  {{NumBlk2|:|<math display="block">F_w= C_d\ \rho  \frac{u^2}{2} S = C_d\ \rho  \frac{u^2}{2} \pi L^2</math>|MHD.3}}

where <math display="inline">C_d</math> is a [[Coefficient of Drag|coefficient of drag]] determined by [[numerical analysis]] and/or simulation,  <math display="inline">\rho u^2/2</math> is the wind pressure, and <math>S=\pi R_{mp}^2</math> is the effective blocking area of the magnetic sail with magnetopause radius <math display="inline">R_{mp} \approx L</math>. Note that this equation has the same form as the [[drag equation]] in [[fluid dynamics]]. <math display="inline">C_d</math> is a function of [[#Coil attack angle effect on thrust and steering angle|coil attack angle on thrust and steering angle]]. The [[Power (physics)|power (W)]] of the plasma wind is the product of velocity and a constant force{{NumBlk2|:|<math display="block">P_w= u \, F_w =  C_d\ \rho  \frac{u^3}{2} \pi R_{mp}^2 = \frac {C_d u \pi}{2 \mu_0 C_{SO} } R_{mp}^2 B_{mp}^2</math>|MHD.4}}

where equation {{EquationNote|MHD.2}} was used to derive the right side.<ref name="Slough2006" />{{Rp|location=Eq (9)}}

==== MHD applicability test ====

As summarized in the [[#Overview|overview]] section, an important condition for a magnetic sail to generate maximum force is that the magnetopause radius be on the order of an ion's radius of gyration. Through analysis, numerical calculation, simulation and experimentation an important condition for a magnetic sail to generate significant force is the MHD applicability test,<ref name=":20">{{Cite journal |last1=Funaki |first1=Ikkoh |last2=Nakayama |first2=Yoshinori |date=2004 |title=Sail Propulsion Using the Solar Wind |url=https://www.jstage.jst.go.jp/article/jsts/20/2/20_2_1/_article |journal=The Journal of Space Technology and Science |volume=20 |issue=2 |pages=2_1–2_16 |doi=10.11230/jsts.20.2_1}}</ref> which states that the standoff distance <math>L</math> must be significantly greater than the ion [[gyroradius]], also called the Larmor radius<ref name=":3" /> or cyclotron radius: {{NumBlk2|:|<math display="block">r_g=\frac {m_i v_\perp}{\mid q \mid B_x C_{Li} } </math>|MHD.5}}

[[File:MHD_applicability_test.jpg|alt=Magnetohydrodynamic (MHD) applicability test|thumb|upright=1.5]]
where ''<math display="inline">m_i</math>'' is the ion mass, <math>v_\perp</math> is the velocity of a particle perpendicular to the magnetic field, <math>|q|</math> is the [[elementary charge]] of the ion, <math display="inline">B_x</math> is the magnetic field flux density at the point of reference <math display="inline">x</math> and <math display="inline">C_{Li}</math> is a constant that differs by source with <math display="inline">C_{Li}=1</math><ref name="Slough2006" /> and <math display="inline">C_{Li}=2</math><ref name=":3" />''.'' For example, in the solar wind with 5 ions/cm<sup>3</sup> at 1 AU with <math>m_i</math> the [[proton mass|proton mass (kg)]], <math>v_\perp = v_{sw}</math> = 400&nbsp;km/s, <math>B_x=B_{mp}</math> = 36 nT with <math>C_{SO}</math>=0.5 from equation {{EquationNote|MHD.2}} at magnetopause and <math display="inline">C_{Li}</math>=2 then <math>r_g \approx</math> 72&nbsp;km.<ref name=":3" />{{Rp|location=Eq (7)}} The MHD applicability test is the ratio <math display="inline">r_g/L</math>. The figure plots <math>C_d</math> on the left vertical axis and lost thrust on the right vertical axis versus the ratio <math display="inline">r_g/L</math>. When <math>r_g/L<1</math>, <math>C_d=3.6</math> is maximum,  at <math display="inline">r_g/L\approx 1</math>, <math display="inline">C_d=2.7</math>, a decrease of 25% from the maximum and at <math display="inline">r_g/L\approx 2</math>,  <math display="inline">C_d\approx 1.5</math>, a 45% decrease. As <math>r_g/L</math> increases beyond one, <math>C_d</math> decreases meaning less thrust from the plasma wind transfers to the spacecraft and is instead lost to the plasma wind. In 2004, Fujita<ref name=":34">{{Cite journal |last=Fujita |first=Kazuhisa |date=2004 |title=Particle Simulation of Moderately-Sized Magnetic Sails |url=https://www.jstage.jst.go.jp/article/jsts/20/2/20_2_26/_article |journal=The Journal of Space Technology and Science |volume=20 |issue=2 |pages=2_26–2_31 |doi=10.11230/jsts.20.2_26}}</ref><ref name=":3" /> published numerical analysis using a hybrid PIC simulation using a magnetic dipole model that treated electrons as a fluid and a kinematic model for ions to estimate the coefficient of drag <math>C_d</math> for a magnetic sail operating in the radial orientation resulting in the following approximate formula:{{NumBlk2|:|<math display="block">C_d(r_g/L) = \begin{cases} 
3.6 \, e^{-0.28 (r_g/L)^2}, & \text {for }  r_g/L <1 \\ 
\frac{3.4}{r_g/L}e^{-0.22 (L/r_g)^2}, & \text {for } r_g/L \ge 1 
\end{cases}
</math>|MHD.6}}The lost thrust is <math>T_{loss}=(1-C_d(r_g/L))/3.6</math>.

==== 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.