Editing Magnetic sail (section)

=== Magnetoplasma sail (MPS) ===
In 2003 Funaki and others proposed an approach similar to the [[#Mini-magnetospheric plasma propulsion (M2P2)|M2P2 design]] and called it the MagnetoPlasma Sail (MPS) that started with a coil <math>R_c</math>=0.2 m and a magnetic field falloff rate of <math>f_o</math>=1.52 with injected plasma creating an effective sail radius of <math>L</math>=26&nbsp;km and assumed a conversion efficiency that transferred a fraction of the solar wind momentum to the spacecraft.<ref name=":30">{{Cite web |last=Funaki |first=I. |date=2003 |title=Study of a Plasma Sail for Future Deep Space Missions |url=http://electricrocket.org/IEPC/0089-0303iepc-full.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://electricrocket.org/IEPC/0089-0303iepc-full.pdf |archive-date=2022-10-09 |access-date=July 7, 2022 |website=electric rocket.org}}</ref><ref>{{Citation |last1=Funaki |first1=Ikkoh |title=Thrust Production Mechanism of a Magnetoplasma Sail |date=2003-06-23 |url=https://arc.aiaa.org/doi/10.2514/6.2003-4292 |work=34th AIAA Plasmadynamics and Lasers Conference |series=Fluid Dynamics and Co-located Conferences |publisher=American Institute of Aeronautics and Astronautics |doi=10.2514/6.2003-4292 |access-date=2022-07-08 |last2=Asahi |first2=Ryusuke |last3=Fujita |first3=Kazuhisa |last4=Yamakawa |first4=Hiroshi |last5=Ogawa |first5=Hiroyuki |last6=Otsu |first6=Hirotaka |last7=Nonaka |first7=Satoshi |last8=Sawai |first8=Shujiro |last9=Kuninaka |first9=Hitoshi|isbn=978-1-62410-096-3 }}</ref> Simulation results indicated a significant increase in magnetosphere size with plasma injection as compared to the Magsail design, which had no plasma injection. Analysis showed how adjustment of the MPS steering angle created force that could reach the outer planets. A satellite trial was proposed. Preliminary performance results were reported but later modified in subsequent papers.

Many MPS papers have been published on the magnetic sail contributing to the understanding of general physical principles of an [[#Artificial magnetospheric model|artificial magnetosphere]], its [[#Magnetohydrodynamic model|magnetohydrodynamic model]], and the design approach for computing the magnetopause distance for a given magnetic field source are documented in the linked sections of this article.

In 2004 Funaki and others analyzed MPS cases where <math>R_c</math>=10 m and <math>R_c</math>=100 m<ref name=":20" /> as summarized in Table 2 predicting a characteristic length <math>L</math> of 50 and 450&nbsp;km producing significant thrust with mass substantially less than the Magsail and hence significant acceleration. This paper detailed the MHD applicability test of equation {{EquationNote|MHD.5}} that the characteristic length must be greater than the ion [[gyroradius]] <math>r_g</math> to effectively transfer solar wind momentum to the spacecraft. In 2005 Yamakawa and others further described a potential trial.<ref>{{Cite journal |last1=Yamakawa |first1=Hiroshi |last2=Funaki |first2=Ikkoh |last3=Nakayama |first3=Yoshinori |last4=Fujita |first4=Kazuhisa |last5=Ogawa |first5=Hiroyuki |last6=Nonaka |first6=Satoshi |last7=Kuninaka |first7=Hitoshi |last8=Sawai |first8=Shujiro |last9=Nishida |first9=Hiroyuki |last10=Asahi |first10=Ryusuke |last11=Otsu |first11=Hirotaka |date=September 2005 |title=Magneto-plasma sail: An engineering satellite concept and its application for outer planet missions |url=https://linkinghub.elsevier.com/retrieve/pii/S0094576505002365 |journal=Acta Astronautica |language=en |volume=59 |issue=8–11 |pages=777–784 |doi=10.1016/j.actaastro.2005.07.003}}</ref>

An analogy with the Earth's magnetosphere and magnetopause in determining the penetration of plasma irregularities into the magnetopause defines the key parameter of a local kinetic plasma beta as the ratio of the dynamic pressure <math>p_{dyn}</math> of the injected plasma over the magnetic pressure <math>p_{dyn}</math> as follows<ref name=":222"/>{{NumBlk2|:|<math>\beta_k = \frac {p_{dyn} }{p_{mag} }= \frac {\rho_{l} u_{l}^2}{B_{l} ^2 /\mu_0}</math>|MPS.1}}where <math>\rho_l</math> kg/m<sup>3</sup> is the local plasma density, <math display="inline">u_l </math> m/s is the local velocity of the plasma and <math>B_l</math> T is the local magnetic field flux density. Simulations have shown that the kinetic beta is smallest near the field source, at magnetopause and the bow shock.<ref name=":222" />

The kinetic <math>\beta_k </math> differs from the [[Plasma beta|thermal plasma beta]] <math display="inline">\beta_i=p_{plasma}/p_{mag}</math> which is the ratio of the plasma thermal pressure to the magnetic pressure, with terms: <math>p_{plasma}=n \, k_B \,T</math> is the plasma pressure with <math>n</math> the number density, <math>k_B</math> the [[Boltzmann constant]] and <math>T</math> the ion temperature; and <math display="inline">p_{mag} = B^2/(2 \mu_0)</math> the magnetic pressure for magnetic field flux density <math>B</math> and <math>\mu_0</math> [[vacuum permeability]]. In the context of the MPS, <math>\beta_k</math> determines the propensity of the injected plasma flow to stretch the magnetic field while <math>\beta_i</math> specifies the relative energy of the injected plasma.<ref>{{Cite web |last=Little |first=Justin M. |date=September 11–15, 2011 |title=Similarity Parameter Evolution within a Magnetic Nozzle with Applications to Laboratory Plasmas |url=http://electricrocket.org/IEPC/IEPC-2011-229.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://electricrocket.org/IEPC/IEPC-2011-229.pdf |archive-date=2022-10-09 |access-date=July 11, 2022 |website=electric rocket.org |publisher=IEPC 2011|publication-place=Wiesbaden, Germany}}</ref>

In 2005 Funaki and others published numerical analysis<ref name=":28">{{Cite web |last=Funaki |first=Ikkoh |date=November 4, 2005 |title=Feasibility Study of Magnetoplasma Sail |url=http://electricrocket.org/IEPC/115.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://electricrocket.org/IEPC/115.pdf |archive-date=2022-10-09 |access-date=July 19, 2022 |website=electricrocket.org |location=Princeton University}}</ref> showing <math>f_0</math>=1.88 for <math>\beta_k</math>=0.1.  In 2009 Kajimura published simulation results<ref>{{Cite journal |last=Kajimura |first=Yoshihiro |date=2009 |title=Numerical Study of Inflation of a Dipolar Magnetic Field in Space by Plasma Jet Injection |url=http://www.jspf.or.jp/JPFRS/PDF/Vol8/jpfrs2009_08-1616.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.jspf.or.jp/JPFRS/PDF/Vol8/jpfrs2009_08-1616.pdf |archive-date=2022-10-09 |journal=J. Plasma Fusion Res. |volume=8 |pages=1616–1621 |via=jspf.or.jp}}</ref> with <math>\beta_k</math>=5 and <math>\beta_i</math> ranging from 6 to 20 that the magnetic field falloff rate <math>f_0</math> with argon and xenon plasma injected into the polar region was <math>f_0</math>=2.1 and with argon plasma injected into the equatorial region was <math>f_0</math>=1.8.

If <math>\beta_k>1</math> then the Injection of a high-velocity, high-density plasma into a magnetosphere as proposed in [[#Mini-magnetospheric plasma propulsion (M2P2)|M2P2]] freezes the motion of a magnetic field into the plasma flow and was believed to inflate the magnetosphere.<ref name=":9" /> However experiments and numerical analysis determined that the solar wind cannot compress the magnetosphere and momentum transfer to the spacecraft is limited since momentum is transferred to injected plasma flowing out of the magnetosphere,<ref name=":222"/> similar to another criticism of M2P2.<ref name=":8" /> 
[[File:Magnetoplasma sail schematic.jpg|thumb|upright=1.5|Magnetoplasma sail (MPS) schematic]]
An alternative is to reduce the plasma injection velocity and density to result in <math>\beta_k<1</math> to achieve a plasma in equilibrium with the inflated magnetic field and therefore induce an equatorial diamagnetic current in the same direction as the coil current as shown in the figure, thereby increasing the magnetic moment of the MPS field source and consequently increasing thrust. In 2013 Funaki and others<ref name=":222"/><ref name=":26">{{Citation |last1=Funaki |first1=Ikkoh |title=Progress in Magnetohydrodynamic and Particle Simulations of Magnetoplasma Sail |url=https://arc.aiaa.org/doi/abs/10.2514/6.2012-4300 |work=48th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit |publisher=American Institute of Aeronautics and Astronautics |doi=10.2514/6.2012-4300 |access-date=2022-07-15 |last2=Kajimura |first2=Yoshihiro |last3=Nishida |first3=Hiroyuki |last4=Ashida |first4=Yasumasa |last5=Yamakawa |first5=Hiroshi |last6=Shinohara |first6=Iku |last7=Yamagiwa |first7=Yoshiki|year=2012 |isbn=978-1-60086-935-8 }}</ref> published simulation and theoretical results regarding how characteristics of the injected plasma affected thrust gain through creation of an equatorial ring current. They defined thrust gain for MPS as <math>G_{MPS}=F_{MPS}/F_{mag}</math>: the ratio of the force generated by low beta plasma injection <math>F_{MPS}</math> divided by that of a pure magnetic sail <math>F_{mag}</math> from equation {{EquationNote|MFM.5}} with <math display="inline">f_o =3</math> and <math>C_{SO}=0.5</math> for <math>r_g\leq L</math> or from equation {{EquationNote|GKM.1}} for <math>r_g>L</math>. They reported <math>G_{MPS}</math> of approximately 40 for magnetospheres less than the MHD applicability test and 3.77 for a larger magnetosphere where MHD applicability occurred, larger than values reported in 2012 of 20 and 3.3, respectively. Simulations revealed that optimum thrust gain occurred for <math>\beta_k<1</math> and <math>\beta_i \approx 10</math>.

In 2014 Arita, Nishida and Funaki published simulation results<ref name=":17" /> indicating that plasma injection created an equatorial ring current and that the plasma injection rate had a significant impact on thrust performance, with the lowest value simulated having the best performance of a thrust gain <math>G_{MPS}</math> of 3.77 with <math>\beta_i \approx 25</math>. They also reported that MPS increased the height of the magnetosphere by a factor of 2.6, which is important since it increases the effective sail blocking area.

In 2014 Ashida and others documented Particle In Cell (PIC) simulation results for a kinematic model for cases where <math>r_g >> L</math> where MHD is not applicable.<ref name=":23">{{Cite journal |last1=Ashida |first1=Yasumasa |last2=Funaki |first2=Ikkoh |last3=Yamakawa |first3=Hiroshi |last4=Usui |first4=Hideyuki |last5=Kajimura |first5=Yoshihiro |last6=Kojima |first6=Hirotsugu |date=2014-01-01 |title=Two-Dimensional Particle-In-Cell Simulation of Magnetic Sails |url=https://arc.aiaa.org/doi/10.2514/1.B34692 |journal=Journal of Propulsion and Power |volume=30 |issue=1 |pages=233–245 |doi=10.2514/1.B34692|hdl=2433/182205 |hdl-access=free }}</ref>  Equation (12) of their study included the additional force of the injected plasma jet <math>F_{jet}</math> consisting of momentum and static pressure of ions and electrons and defined thrust gain as <math display="inline">F_{MPS}/(F_{mag}+F_{jet})</math>, which differs from the definition of a term by the same name in other studies.<ref name=":222"/><ref name=":26" />  It represents the gain of MPS over that of simply adding the magnetic sail force and the plasma injection jet force. For the values cited in the conclusion, <math>F_{MPS}/F_{mag}</math> is 7.5 in the radial orientation.
[[File:Summary of MPS thrust gain results.jpg|thumb|upright=1.8|Summary of MPS thrust gain results]]
Since a number of results were published by different authors at different times, the figure summarizes the reported thrust gain <math>G_{MPS}</math> versus magnetosphere size (or characteristic length <math>L</math>) with the source indicated in the legend as follows for simulation results Arita14,<ref name=":17" /> Ashida14,<ref name=":23" /> Funaki13,<ref name=":222"/> and Kajimura10.<ref>{{Citation |last1=Kajimura |first1=Yoshihiro |title=Thrust Evaluation of Magneto Plasma Sail by Using Three-Dimensional Hybrid PIC Code. |url=https://arc.aiaa.org/doi/abs/10.2514/6.2010-6686 |work=46th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit |publication-date=July 25–28, 2010 |publisher=American Institute of Aeronautics and Astronautics |doi=10.2514/6.2010-6686 |access-date=2022-07-19 |last2=Funaki |first2=Ikkoh |last3=Shinohara |first3=Iku |last4=Usui |first4=Hideyuki |last5=Yamakawa |first5=Hiroshi|year=2010 |isbn=978-1-60086-958-7 |s2cid=124976334 }}</ref> Simulation results require significant compute time, for example it took 1024 CPUs 4 days to simulate the simplest case and 4096 CPUs one week to simulate a more complex case.<ref name=":25" /> A thrust gain between 2 and 10 is common with the larger gains with a magnetic nozzle injecting plasma in one direction in opposition to the solar wind.<ref name=":16" /><ref name=":24">{{Cite web |last=Kajimura |first=Yohihiro |date=July 4–10, 2015 |title=Thrust Performance of Magneto Plasma Sail with a Magnetic Nozzle |url=http://electricrocket.org/IEPC/IEPC-2015-329_ISTS-2015-b-329.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://electricrocket.org/IEPC/IEPC-2015-329_ISTS-2015-b-329.pdf |archive-date=2022-10-09 |access-date=July 11, 2022 |website=electric rocket.org |publisher=IEPC 2015|publication-place=Hyogo-Kobe, Japan}}</ref> The MHD applicability test of equation {{EquationNote|MHD.5}} for the solar wind is <math>L \approx</math>72&nbsp;km. Therefore, the estimated force of the MPS is that of equation {{EquationNote|MHD.3}} multiplied by the empirically determined thrust gain <math>G_{MPS}</math> from the figure multiplied by the percentage thrust loss <math>T_{loss}</math> from equation {{EquationNote|MHD.6}}

{{NumBlk2|:|<math>F_{MPS}=  G_{MPS} \, (1-T_{loss}) \, C_d\ \rho  \frac{u^2}{2} \pi L^2, \, \, r_g \le L</math>|MPS.2}}

For example, using solar wind parameters <math>\rho</math>=8x10<sup>−21</sup> kg/m<sup>3</sup> and <math>u</math>=500&nbsp;km/s then <math>r_g</math>=72&nbsp;km and <math>B_{mp}</math>=4x10<sup>−8</sup> T.  With <math>L</math>=10<sup>5</sup> m for <math>f_o</math>=3 then <math>r_g < L</math>  and <math>T_{loss}\approx</math> 11% from equation {{EquationNote|MHD.6}}. The magnetic field only force with a coil radius of <math>R_c</math>=6,300 m and coil current <math>I_c</math>=1.6x10<sup>6</sup> A yields <math>B_0</math>=1.6x10<sup>−4</sup> T from equation {{EquationNote|MFM.2}} and with <math>C_d</math>=5 the magnetic force only is 175 N from equation {{EquationNote|MFM.5}}. Determining <math>G_{MPS} \approx</math>4 from the figure at <math>L</math>=10<sup>5</sup> m as the multiplier for the magnetic-only force then the MPS force <math>F_{MPS} \approx</math>700 N.

Since MPS injects ionized gas at a rate of  <math>m_{in}</math> that can be viewed as a propellant it has a ''[[specific impulse]]'' <math>I_{sp}=F_{MPS}/m_{in}/g_0</math> where <math>g_0</math> is the acceleration of [[Earth's gravity]].  Funaki<ref name=":222" /> and Arita<ref name=":17" /> stated <math>m_{in}</math>=0.31&nbsp;kg/day. Therefore <math>I_{sp}</math>=28,325 s per newton of thrust force. The equivalent exhaust velocity <math>v_e=g_0 \, I_{sp}</math> is 278&nbsp;km/s per newton of thrust force.

In 2015 Kajimura and others published simulation results for thrust performance<ref name=":24" /> with plasma injected by a magnetic nozzle, a technology used in [[Variable Specific Impulse Magnetoplasma Rocket|VASIMR]]. They reported a thrust gain <math>G_{MPS}</math> of 24 when the ion [[gyroradius]] <math>r_g </math> (see equation {{EquationNote|MHD.5}}) was comparable to the characteristic length <math>L </math>, at the boundary of the [[#MHD applicability test|MHD applicability test]]. The optimal result occurred with a thermal <math>\beta_i \approx 1</math> with some decrease for higher values of thermal beta.

In 2015 Hagiwara and Kajimura published experimental thrust performance test results with plasma injection using a [[magnetoplasmadynamic thruster]] (aka MPD thruster or MPD Arcjet) in a single direction opposite the solar wind direction and a coil with the axial orientation.<ref name=":16" /><ref name=":24" /> This meant that <math display="inline">F_{jet}</math> provided additional propulsive force. Density plots explicitly show the increased plasma density upwind of the bow shock originating from the MPD thruster. They reported that <math display="inline">F_{MPS}>>F_{mag}+F_{jet}</math> showing how MPS inflated the magnetic field to create more thrust than the magnetic sail alone plus that of the <<text gap here>>. The conclusion of the experiment was that the thrust gain <math>G_{MPS}</math> was approximately 12 for a scaled characteristic length of <math>L </math> = 60&nbsp;km. In the above figure, note the significant improvement in thrust gain at <math>L </math> = 60&nbsp;km.as compared with only plasma injection.

In this example, using solar wind parameters <math>\rho</math>=8x10<sup>−21</sup> kg/m<sup>3</sup> and <math>u</math>=500&nbsp;km/s then <math>r_g</math>=72&nbsp;km and <math>B_{mp}</math>=4x10<sup>−8</sup> T.  With <math>L</math>=60&nbsp;km for <math>f_o</math>=3 then <math>r_g \approx L</math>  and <math>T_{loss}\approx</math> 28% from equation {{EquationNote|MHD.6}}. The magnetic field only force with a coil radius of <math>R_c</math>=2,900 m and coil current <math>I_c</math>=1.6x10<sup>6</sup> A yields <math>B_0</math>=3.5x10<sup>−4</sup> T from equation {{EquationNote|MFM.2}} and with <math>C_d</math>=5 the magnetic force only is 51 N from equation {{EquationNote|MFM.5}}. Given <math>G_{MPS}</math>=12 as the multiplier for the magnetic only force then the MPS force <math>F_{MPS} \approx</math>611 N.

In 2017 Ueno published a design proposing use of multiple coils to generate a more complex magnetic field to increase thrust production.<ref>{{Cite web |last=Ueno |first=Kazuma |date=2017 |title=Multi-Coil Magnetic Sail Experiment in Laboratory |url=http://www.ea.u-tokai.ac.jp/horisawa/1ssets/9.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.ea.u-tokai.ac.jp/horisawa/1ssets/9.pdf |archive-date=2022-10-09 |access-date=July 11, 2022 |website=www.ea.u-tokai.ac.jp}}</ref>  In 2020 Murayama and others published additional experimental results for a multi-pole MPD thruster.<ref name="Murayama" /> In 2017 Djojodihardjo published a conceptual design using MPS for a small (~500&nbsp;kg)  Earth observation satellite.<ref>{{Cite journal |last=Djojodihardjo |first=Harijono |date=August 2017 |title=ANALYSIS OF CONCEPTUAL MAGNETOSPHERIC PLASMA PROPULSION FOR SMALL EARTH OBSERVATION SATELLITE |url=https://www.academia.edu/34985891 |journal=1st IAA North East Asia Symposium on Small Satellites |location=Ulaanbaatar, Mongolia |volume=1 |via=Academia.edu}}</ref>

In 2020 Peng and others<ref>{{Cite book |last1=Peng |first1=Zhong |last2=Peng |first2=Yuchuan |last3=Ding |first3=Liang |last4=Li |first4=Hao |last5=Zhao |first5=Hua |last6=Li |first6=Tao |last7=Zong |first7=Yi |title=Signal and Information Processing, Networking and Computers |chapter=Global MHD Simulation of the Magnetic Sail Expansion by Plasma Injection |date=2020 |editor-last=Wang |editor-first=Yue |editor2-last=Fu |editor2-first=Meixia |editor3-last=Xu |editor3-first=Lexi |editor4-last=Zou |editor4-first=Jiaqi |chapter-url=https://link.springer.com/chapter/10.1007/978-981-15-4163-6_23 |series=Lecture Notes in Electrical Engineering |volume=628 |language=en |location=Singapore |publisher=Springer |pages=190–197 |doi=10.1007/978-981-15-4163-6_23 |isbn=978-981-15-4163-6|s2cid=216501435 }}</ref> published MHD simulation results for a magnetic dipole with plasma injection operating in [[Low Earth orbit]] at 500&nbsp;km within the Earth's [[Ionosphere]] where the ion number density is approximately 10<sup>11</sup> m<sup>−3</sup>. As reported in Figure 3, the magnetic field strength initially falls off as 1/r and then approaches 1/r<sup>2</sup> at larger distances from the dipole. The radius of the artificial mini-magnetosphere could extend up to 200 m for this scenario. They reported that the injected plasma reduced magnetic field fall off rate and created of a drift current, similar to earlier reported MPS results for the solar wind.<ref name=":23" />