Editing Magnetic sail (section)

== Proposed magnetic sail systems ==
This section contains a subsection for each of the proposed magnetic sail designs introduced in the summary. Each subsection begins with a high-level description of that design and an illustration. The cited references are technical and contain many equations, for which this article includes where applicable a common notation described in the [[#Physical principles|Physical principles]] section, and in other cases the notation from a cited reference. The focus is to include equations used in the [[#Performance comparison|Performance comparison]] section. The subsections include plots of variables with relevant units related to this objective that are preceded by a summary description.

=== Magsail (MS) ===
Andrews was working on use of a magnetic scoop to gather [[interstellar medium|interstellar material]]  as propellant for a nuclear electric [[Ion thruster|ion drive]] spacecraft, allowing the craft to operate in a similar manner to a [[Bussard ramjet]], whose history goes back to at least 1973.<ref>{{Cite journal |last=Jackson |first=Albert |date=2016 |title=Three Interstellar Ramjets |url=http://large.stanford.edu/courses/2021/ph241/mccullough1/docs/jackson-2016.pdf |journal=Tviw 2016}}</ref> Andrews asked Zubrin to help compute the magnetic scoop drag against the interplanetary medium, which turned out to be much greater than the ion drive thrust.  The ion drive component of the system was dropped, and use of the concept of using the magnetic scoop as a magnetic sail or [[#Magsail (MS)|Magsail (MS)]] was born.<ref>{{Cite book |last=Andrews |first=Dana |title=Chasing the Dream |publisher=Classic Day Publishing |year=2020 |isbn=9781598492811}}</ref>[[File:Andrews_Zubrin_Magsail.jpg|thumb|upright=1.5|Andrews & Zubrin Magsail]]
The figure shows the magsail design<ref name=":13" /> consisting of a loop of [[superconductor|superconducting]] wire of radius <math>R_c</math> on the order of 100&nbsp;km that carries a direct current <math>I_c</math> that generates a magnetic field, which was modeled according to the [[Biot–Savart law]] inside the loop and as a [[magnetic dipole]] far outside the loop. With respect to the plasma wind direction a magsail may have a radial (or normal) orientation or an axial orientation that can be adjusted to provide torque for steering.  In non-axial configurations lift is generated that can change the spacecraft's momentum. The loop connects via shroud lines (or tethers) to the spacecraft in the center. Because a loop carrying current is forced outwards towards a circular shape by its magnetic field, the sail could be deployed by unspooling the conductor wire and applying a current through it via the peripheral platforms.<ref name=":1" /> The loop must be adequately attached to the spacecraft in order to transfer momentum from the plasma wind and would pull the spacecraft behind it as shown in the axial configuration in the right side of the figure. This design has a significant advantage of requiring no propellant and is thus a form of [[Field propulsion#Practical methods|field propulsion]] that can operate indefinitely.<ref name=":4" />{{Rp|location=Sec VIII}}

==== MHD model ====
Analysis of magsail performance was done using a simulation and a fluid (i.e., MHD) model with similar results observed for one case.<ref name=":13" /> The [[Magnetic moment#Amperian loop model|magnetic moment of a current loop (A m<sup>2</sup>)]] is <math>\mathbf m= I_c \pi R_c^2</math>  for a current of <math display="inline">I_c</math> A and a loop of radius <math display="inline">R_c</math> m.  Close to the loop, the magnetic field at a distance <math>z</math> along the center-line axis perpendicular to the loop is derived from the [[Biot-Savart law]] as follows.<ref>{{Cite web |last=Zhan |first=Marcus |date=2003 |title=Electromagnetic Field Theory: A Problem Solving Approach |url=https://ocw.mit.edu/courses/res-6-002-electromagnetic-field-theory-a-problem-solving-approach-spring-2008/pages/textbook-contents/ |access-date=July 3, 2022 |website=cow.mit.edu}}</ref>{{Rp|location=sec 5-2, Eq (25)}}

{{NumBlk2|:|<math display="block">B_{cl} (z)=\frac {\mu_0 I_c R_c^2}{2(z^2 +R_c^2)^{3/2} }</math>|MS.1}}

At a distance far from the loop center the magnetic field is approximately that produced by a [[Magnetic dipole#External magnetic field produced by a dipole moment|magnetic dipole]]. Te pressure at the magnetospheric boundary is doubled due to compression of the magnetic field and stated by the following equation at a point along the center-line axis or the target magnetopause standoff distance <math>L_Z</math>.<ref name=":13" />{{Rp|location=Eq (5)}}{{NumBlk2|:|<math>p_{mb}=\frac {B_{cl}(0)^2}{2 \mu_0} \Biggl(\frac {R_c}{L_Z}\Biggr)^6</math>|MS.2}}
Equating this to the dynamic pressure for a plasma environment <math display="inline">p_{mb}=\rho \, u_{pe}^2 /2</math>, inserting <math display="inline">B_{cl}(0)</math> from equation {{EquationNote|MS.1}} and solving for <math>L_Z</math> yields<ref name=":13" />{{Rp|location=Eq (6)}}
{{Numbered block 2|:|<math>L_Z=1.26 \, C_Z, where \, \, C_Z = R_c \Biggl( \frac {B_{cl}(0)}{u_{pe} \sqrt{\rho \mu_0} } \Biggr)^{1/3}</math>|MS.3}}

Andrews and Zubrin derived the drag (thrust) force of the sail <math>F_D</math><ref name=":13" />{{Rp|location=Eq (8)}} that determined the characteristic length <math>L_Z</math> for a tilt angle, but according to Freeland<ref name=":14" />{{Rp|location=Sec 6.5}} an error was made in numerical integration in choosing the ellipse downstream from the magnetopause instead of the ellipse upstream that made those results optimistic by a factor of approximately 3.1, which should be used to correct any drag(thrust) force results using<ref name=":13" />{{Rp|location=Eq 8}} Instead, this article uses the approximation<ref name=":14" />{{Rp|location=Eq (108)}}  for a spherical bubble that corrects this error and is close to the analytical formula for the axial configuration as the force for the Magsail as follows{{NumBlk2|:|<math>F_{MS} = 1.214 \, \,  \frac{\rho u_{pe}^2}{2} \pi \, L_Z^2</math>|MS.4}}
In 2004 Toivanen and Janhunen did further analysis on the Magsail that they called a Plasma Free MagnetoPause (PFMP) that produced similar results to that of Andrews and Zubrin.<ref name=":8" />

==== Coil mass and current (CMC) ====

The minimum required mass to carry the current in equation {{EquationNote|MS.1}} or other magnetic sail designs from Andres/Zubrin (9)<ref name=":13" />{{Rp|location=Eq (9)}} and Crowl<ref name=":29">{{Cite web |last=Crowl |first=Adam |date=September 2017 |title=High-Speed Magnetic-Sail Interstellar Precursor Missions Enabled by Metastable Metallic Hydrogen |url=https://www.researchgate.net/publication/321110213 |access-date=August 14, 2022 |work=68th International Astronomical Conference |location=Adelaide Australia}}</ref>{{Rp|location=Eq (3)}} as follows:{{NumBlk2|:|<math>_{min}M_c \, = 2 \, \pi \, R_c \, I_c \, / (J _e/ \delta_c)</math>|CMC.1}}

where <math>J_e</math> is the superconductor critical [[Current density|current density (A/m<sup>2</sup>)]] and <math display="inline">\delta_c</math> is the coil material density, for example <math>J_e</math> = 1x10<sup>11</sup> A/m<sup>2</sup> and <math display="inline">\delta_c</math> = 6,500&nbsp;kg/m<sup>3</sup> for a superconductor in Freeland<ref name=":14" />{{Rp|location=Apdx A}} The physical mass of the coil is

{{NumBlk2|:|<math>_{phy}M_c= (2 \, \pi  +N_{tether} ) \, R_c  \, \pi \, r_c^2 \, \delta_c</math>|CMC.2}}

where <math>r_c </math> is the radius of the superconductor wire, for example that necessary to handle the tension for a particular use case, such as deceleration in the ISM where  <math>r_c </math> = 10&nbsp;mm.<ref name=":14" />{{Rp|location=Apdx A}}  The <math>N_{tether}</math> factor (e.g., 3) accounts for mass of the tether (or shroud) lines to connect the coil to a spacecraft. Note that <math>_{phy} M_c</math> with <math>N_{tether}</math>=0 must be no less than <math>_{min}M_c</math> in order for the coil to carry the [[Superconductivity#Elementary properties of superconductors|superconductor critical current]] <math>I_{cc}=J_e \pi r_c^2</math> amperes for a coil wire of radius <math>r_c</math>, for example <math>I_{cc}</math> = 7,854 [[Kiloampere|kiloampere (kA.)]]<ref name=":14" />{{Rp|location=Apdx A}}

Setting equation {{EquationNote|CMC.2}} with <math>N_{tether}</math>=0 equal to equation {{EquationNote|CMC.1}} and solving for <math>r_c</math> yields the minimum required coil radius

{{NumBlk2|:|<math>_{min} r_c =\sqrt{I_c/(J_e \pi)}
</math>|CMC.3}}

If operated within the solar system, high temperature [[superconducting wire]] (HTS) is necessary to make the magsail practical since required current is large, millions of amperes. Protection from solar heating is necessary closer to the Sun, for example by highly reflective coatings.<ref>{{Cite journal |last1=Youngquist |first1=Robert C. |last2=Nurge |first2=Mark A. |last3=Johnson |first3=Wesley L. |last4=Gibson |first4=Tracy L. |last5=Surma |first5=Jan M. |date=2018-05-01 |title=Cryogenic Deep Space Thermal Control Coating |url=https://arc.aiaa.org/doi/10.2514/1.A34019 |journal=Journal of Spacecraft and Rockets |volume=55 |issue=3 |pages=622–631 |doi=10.2514/1.A34019 |bibcode=2018JSpRo..55..622Y |issn=0022-4650}}</ref> If operated in interstellar space low temperature superconductors (LTS) could be adequate since the temperature of a vacuum is 2.7{{nbsp}}[[Kelvin|Kelvins (K)]], but radiation and other heat sources from the spacecraft may render LTS impractical. The critical current <math>I_{cc}</math> of the [[Superconducting wire#Coated superconductor tape or wire|HTS YBCO coated superconductor wire]] increases at lower temperatures with a current density <math>J_e</math> of 6x10<sup>10</sup> A/m<sup>2</sup> at 77{{nbsp}}K and 9x10<sup>11</sup> A/m<sup>2</sup> at 5{{nbsp}}K.

==== Magsail kinematic model (MKM) ====
The MHD applicability test of equation {{EquationNote|MHD.5}} fails in some ISM deceleration cases and a kinematic model is necessary, such as the one documented in 2017 by [[Claudius Gros]] summarized here.<ref name="gros2017">{{Cite journal |last=Gros |first=Claudius |date=2017 |title=Universal scaling relation for magnetic sails: Momentum braking in the limit of dilute interstellar media |journal=Journal of Physics Communications |volume=1 |issue=4 |page=045007 |arxiv=1707.02801 |bibcode=2017JPhCo...1d5007G |doi=10.1088/2399-6528/aa927e |s2cid=119239510}}</ref> A spacecraft with an overall mass <math>m_{tot}</math>  and velocity <math>v</math> follows<ref name="gros2017" />{{Rp|location=Eq (1)}} of motion as:

{{NumBlk2|:|<math>F_{MKM} = m_{tot}\dot{v} = 
-\left(A_G(v)n_pv\right)(2m_pv)
= 2 \, \rho_{im} \, v^2 A_G(v),</math>|MKM.1}}where <math>F_{MKM}</math> N is force predicted by this model,  <math>n_p</math> m<sup>−3</sup> is the proton number density, <math>m_p</math> kg is the [[proton]] mass, <math>\rho = m_p n_p</math> kg/m<sup>3</sup> the plasma density, and <math>A_G(v)</math> m<sup>2</sup> the effective reflection area. This equation assumes that the spacecraft encounters <math>A_G(v) n_p v</math> particles per second and that every particle of mass <math>m_p</math> is completed reflected. Note that this equation is of the same form as {{EquationNote|MFM.5}} with <math>C_d</math>=4, interpreting the <math>C_d</math> term as just a number.

Gros numerically determined the effective reflection area <math display="inline">A_G(v)</math> by integrating the degree of reflection of approaching protons interacting with the superconducting loop magnetic field according to the [[Biot–Savart law|Biot-Savart law]]. The reported result was independent of the loop radius <math>R_c</math>. An accurate curve fit as reported in Figure 4 to the numerical evaluation for the effective reflection area for a magnetic sail in the axial configuration from equation (8) was

{{NumBlk2|:|<math>
A_G(v) = 0.081\pi R_c^2\left[ \log
\left( \frac{cI}{v I_G } \right) \right]^3 \, \, , v/c<  I/I_G
</math>
|MKM.2}}

where <math>\pi R_c^2</math> is the area enclosed by the current carrying loop, <math>c</math> the [[speed of light]], and the value  <math>I_G=1.55\cdot10^{6}</math> A determined a good curve fit for <math>I</math>=10<sup>5</sup>  A, the current through the loop. In 2020, Perakis published an analysis<ref name=":332">{{Cite journal |last=Perakis |first=Nikolaos |date=December 2020 |title=Maneuvering through solar wind using magnetic sails |url=https://linkinghub.elsevier.com/retrieve/pii/S0094576520304471 |journal=Acta Astronautica |language=en |volume=177 |pages=122–132 |doi=10.1016/j.actaastro.2020.07.029|bibcode=2020AcAau.177..122P |s2cid=224882966 }}</ref> that corroborated the above formula with parameters selected for the solar wind and reported a force no more than 9% less than the Gros model for <math>I</math>=10<sup>5</sup> A and <math>R_c</math>=100 m with the coil in an axial orientation.. That analysis also reported on the effect of magsail tilt angle on lift and side forces for a use case in maneuvering within the solar system.

For comparison purposes, the effective sail area determined for the magsail by Zubrin from equation {{EquationNote|MS.3}} with the 3.1 correction factor from Freeland applied and using the same velocity value (resolving the discrepancy noted by Gros) as follows:

{{NumBlk2|:|<math>
A_Z(v) = \frac {1.124 \times 1.26} {3.1} \,L_Z
</math>
|MKM.3}}
[[File:Magsail MHD and kinematic model effective sail area.jpg|thumb|upright=1.5|Magsail MHD and kinematic model effective sail area]]
The figure shows the normalized effective sail area normalized by the coil area <math>\pi R_c^2</math> for the MKM case from Gros of equation {{EquationNote|MKM.1}} and for Zubrin from equation {{EquationNote|MKM.3}} for <math>I \approx I_G</math>,  <math>R_c</math>=100&nbsp;km, and <math>n_p</math>=0.1&nbsp;cm<sup>−3</sup> for the [[G-Cloud|G-cloud]] on approach to Alpha Centauri corresponding to ISM density <math>\rho_{im}=1.67\times10^{-22}</math> kg/m<sup>3</sup> consistent with that from Freeland<ref name=":14" /> plotted versus the spacecraft velocity relative to the speed of light <math display="inline">\beta =v/c</math>.  A good fit occurs for these parameters, but for different values of <math>R_c</math> and <math>I</math> the fit can vary significantly. Also plotted is the MHD applicability test of ion gyroradius divided by magnetopause radius <math>r_g/R_{mp}</math> <1 from equation {{EquationNote|MHD.4}} on the secondary axis. Note that MHD applicability occurs at <math>v/c</math> < 1%. For comparison, the 2004 Fujita <math>C_d</math> as a function of <math>r_g/R_{mp}</math> from the [[#MHD applicability test|MHD applicability test]] section is also plotted. Note that the Gros model predicts a more rapid decrease in effective area than this model at higher velocities. The normalized values of <math>A_G(v)</math> and <math>A_Z(v)</math> track closely until <math>\beta = v/c \approx</math> 10% after which point the Zubrin magsail model of Equation {{EquationNote|MS.4}} becomes increasingly optimistic and equation {{EquationNote|MKM.2}} is applicable instead. Since the models track closely up to  <math>\beta \approx</math> 10%, with the kinematic model underestimating effective sail area for smaller values of <math>\beta</math> (hence underestimating force), equation {{EquationNote|MKM.1}} is an approximation for both the MHD and kinematic region. The Gros model is pessimistic for <math>\beta</math> < 0.1%.

Gros used the analytic expression for the effective reflection area <math>A_G(v)</math> from equation {{EquationNote|MKM.3}} for explicit solution for the required distance <math>x_f</math> m to decelerate to final velocity <math>v_f \approx 0.013 c</math> m/s from<ref name="gros2017" />{{Rp|location=Eq (10)}} given an initial velocity <math>v_0</math> m/s for a spacecraft mass <math>m_{tot}</math> kg as follows: {{NumBlk2|:|<math>
x_{f}=
\frac {m_{tot} \, [g(v_0)-g(v_f)] }{0,081 \, m_p \, n_p \pi \, R_c^2}
</math>|MKM.4}}

where <math display="inline">g(v)=ln^{-2}(\frac {v \, I_G}{c \, I})</math>. When <math>v_f</math>=0 the above equation is defined in<ref name="gros2017" />{{Rp|location=Eq (11)}} as <math>x_{max}</math>, which enabled a closed form solution of the velocity at a distance <math>x, v(x)</math> in<ref name="gros2017" />{{Rp|location=Eq (12)}} with numerical integration required to compute the time required to decelerate.<ref name="gros2017" />{{Rp|location=Eq (14)}} Equation (16) The optimal current that minimized <math>
x_{max}
</math> as <math>I_{opt}=e \beta_0 I_G</math> where <math>\beta_0=v_0/c</math>.<ref name="gros2017" />{{Rp|location=Eq (16)}} In 2017 Crowl<ref name=":29" /> optimized coil current for the ratio of effective area <math>A(v)</math> over total mass <math>m_{tot}</math> and derived the result <math>I_{opt}=e^3 \, \beta_0 I_G</math>.<ref name="gros2017" />{{Rp|location=Eq (15)}} That paper used results from Gros for the stopping distance <math>
x_{max}
</math> and time to decelerate.
[[File:Magsail ISM deceleration distance and time comparison.jpg|thumb|upright=1.5|Magsail ISM deceleration distance and time comparison]]
The figure plots the distance traveled while decelerating <math>x_d</math> and time required to decelerate <math>t_d</math> given a starting relative velocity <math>\beta_0 = v_0/c</math> and a final velocity <math>v_f=0.013 c</math> m/s consistent with that from Freeland<ref name=":14" /> for the same parameters above. Equation {{EquationNote|CMC.1}} gives the magsail mass <math>M_s</math> as 97 tonnes assuming payload mass <math>M_p</math> of 100 tonnes using the same values used by Freeland<ref name=":14" /> of  <math>J_e</math> = 10<sup>11</sup> A/m<sup>2</sup> and <math>\delta_c
</math>=6,500&nbsp;kg/m<sup>3</sup> for the superconducting coil. Equation {{EquationNote|MS.4}} gives Force for the magsail multiplied by <math>C_d</math>=4 for the Andrews/Zubrin model to align with equation {{EquationNote|MHD.3}} definition of force from the Gros model. [[Acceleration]] is force divided by mass, [[velocity]] is the integral of acceleration over the deceleration time interval <math>t_d</math> and deceleration distance traveled <math>x_d</math> is the integral of the velocity over <math>t_d</math>. Numerical integration resulted in the lines plotted in the figure with deceleration distance traveled plotted on the primary vertical axis on the left and time required to decelerate on the secondary vertical axis on the right. Note that the MHD Zubrin model and the Gros kinematic model predict nearly identical values of deceleration distance up to <math>\beta_0</math>~ 5% of c, with the Zubrin model predicting less deceleration distance and shorter deceleration time at greater values of <math>\beta_0</math>. This is consistent with the Gros model predicting a smaller effective area <math>A(v)</math> at larger values of <math>\beta_0</math>. The value of the closed form solution for deceleration distance <math>
x_f
</math> from {{EquationNote|MKM.4}} for the same parameters closely tracks the numerical integration result.

==== Specific designs and mission profiles ====
Dana Andrews and [[Robert Zubrin]] first proposed the magnetic sail concept in 1988 for interstellar travel for acceleration by a fusion rocket, coasting and then deceleration via a magsail at the destination that could reduce flight times by 40–50 years<ref name=":6" /> In 1989 details for interplanetary travel were reported<ref name=":13" /> In 1990 Andrews and Zubrin reported on an example for [[#Acceleration/deceleration in a stellar plasma wind|solar wind parameters]] one [[Astronomical unit|AU]] away from the Sun, with <math display="inline">n_i = 5 \times 10^{6}</math> m<sup>−3</sup> with only protons as ions, apparent wind velocity <math display="inline">u_{sw}</math>=500&nbsp;km/s the field strength required to resist the [[dynamic pressure]] of the solar wind is 50 nT from equation {{EquationNote|MHD.2}}. With radius <math>R_c</math>=100&nbsp;km and magnetospheric bubble of <math display="inline">L </math>={{convert|500|km|mi|adj=||abbr=on}} reported a thrust of 1980 newtons and a coil mass of 500 tonnes.<ref name=":4" />  For the above parameters with the correction factor of 3.1 applied to equation {{EquationNote|MS.4}} yields the same thrust and equation {{EquationNote|CMC.1}} yields the same coil mass. Results for another 4 solar wind cases were reported,<ref name=":13" /> but the MHD applicability test of equation {{EquationNote|MHD.5}} fails in these cases.

In 2015, Freeland documented a use case with acceleration away from Earth by a fusion drive with a magsail used for interstellar deceleration on approach to [[Alpha Centauri|Alpha Centaturi]] as part of a study to update [[Project Icarus (interstellar)|Project Icarus]]<ref name=":14" /> with <math>R_c</math>=260&nbsp;km, an initial <math display="inline">R_{mp}</math> of 1,320&nbsp;km and ISM density <math>\rho_{im}=1.67\times10^{-22}</math> kg/m<sup>3</sup>, almost identical to the n(H I) measurement of 0.098&nbsp;cm<sup>−3</sup> by Gry in 2014.<ref name="Gry A58">{{Cite journal |last1=Gry |first1=Cécile |last2=Jenkins |first2=Edward B. |date=July 2014 |title=The interstellar cloud surrounding the Sun: a new perspective |url=http://www.aanda.org/10.1051/0004-6361/201323342 |journal=Astronomy & Astrophysics |volume=567 |pages=A58 |arxiv=1404.0326 |bibcode=2014A&A...567A..58G |doi=10.1051/0004-6361/201323342 |issn=0004-6361 |s2cid=118571335}}</ref>  The Freeland study predicted deceleration from 5% of light speed in approximately 19 years.  The coil parameters <math>J_e</math>=10<sup>11</sup> A/m<sup>2</sup>, <math>r_{sc}</math>= 5&nbsp;mm, <math>\rho_{sc}</math>=6,500&nbsp;kg/m<sup>3</sup>, resulted in an estimated coil mass of <math>M_c(phy)</math>=1,232 tonnes. Although the critical current density  <math>J_e</math> was based upon a 2000 Zubrin NIAC report projecting values through 2020, the assumed value is close to that for commercially produced [[Superconducting wire#Coated superconductor tape or wire|YBCO coated superconductor wire]] in 2020. The mass estimate may be optimistic since it assumed that the entire coil carrying mass is superconducting while 2020 manufacturing techniques place a thin film on a non-superconducting substrate. For the interstellar medium plasma density <math>\rho_{im}</math>=1.67x10<sup>−22</sup>  with an apparent wind velocity 5% of light speed, the ion gyroradius is 570&nbsp;km and thus the design value for <math display="inline">R_{mp}</math> meets the MHD applicability test of equation {{EquationNote|MHD.5}}. Equation {{EquationNote|MFM.3}} gives the required coil current as <math>I_c</math>~7,800 kA and from equation {{EquationNote|CMC.1}} <math>M_c(min)</math>= 338 tonnes; however, but the corresponding superconducting wire minimum radius from equation {{EquationNote|CMC.3}}  is <math>r_c(min) </math> =1&nbsp;mm, which would be insufficient to handle the decelerating thrust force of <math>F_{MS}</math> ~ 100,000 N predicted by equation {{EquationNote|MS.4}} and hence the design specified <math>r_{sc}</math>= 5&nbsp;mm to meet structural requirements. In a complete design, the mass of infrastructure, including coil shielding to maintain critical temperature and survive abrasion in outer space, must also be included. Appendix A estimates these as 90 tonnes for wire shielding and 50 tonnes for the spools and other magsail infrastructure. Freeland compared this magsail deceleration design with one where both acceleration and deceleration were performed by a fusion engine and reported that the mass of such a "dirty Icarus" design was over twice that of a magsail used for deceleration. <!-- Cite Crowl analysis of Fusion versus magsail for deceleration? -->An Icarus design published in 2020 used a [[Z-pinch]] fusion drive in an approach called Firefly that dramatically reduced mass of the fusion drive and made fusion only drive performance for acceleration and deceleration comparable to the fusion for acceleration and magsail for deceleration design.<ref>{{Cite journal |last1=Swinney |first1=R.W. |last2=Freeland II |first2=R.M. |last3=Lamontagne |first3=M. |date=2020-04-10 |title=Project Icarus: Designing a Fusion Powered Interstellar Probe |journal=Acta Futura|issue=12 |pages=47–59 |url=https://zenodo.org/record/3747274 |doi=10.5281/ZENODO.3747274}}</ref><!-- Need to confirm that a source states this:  "although it could be used to decelerate in other directions, such as on approach to a star."  Magsail Freeland describes detaching magsail and using fusion for such deceleration. -->

In 2017, Gros<ref name="gros2017" /> reported numerical examples for the [[#Magsail kinematic model (MKM)|Magsail kinematic model]] that used different parameters and coil mass models than those used by Freeland. That paper assumed hydrogen ion (H I) number densities of 0.05-0.2&nbsp;cm<sup>−3</sup> (9x10<sup>−23</sup> - 3x10<sup>−22</sup> kg/m<sup>3</sup>) for the warm local clouds<ref>{{Cite journal |last1=Frisch |first1=Priscilla C. |last2=Redfield |first2=Seth |last3=Slavin |first3=Jonathan D. |date=2011-09-22 |title=The Interstellar Medium Surrounding the Sun |url=https://www.annualreviews.org/doi/10.1146/annurev-astro-081710-102613 |journal=Annual Review of Astronomy and Astrophysics |language=en |volume=49 |issue=1 |pages=237–279 |doi=10.1146/annurev-astro-081710-102613 |bibcode=2011ARA&A..49..237F |issn=0066-4146}}</ref> and about 0.005&nbsp;cm<sup>−3</sup> (9x10<sup>−23</sup> kg/m<sup>3</sup>)  for voids of the local bubble.<ref>{{Cite journal |last1=Welsh |first1=Barry Y. |last2=Shelton |first2=Robin L. |date=September 2009 |title=The trouble with the Local Bubble |journal=Astrophysics and Space Science |language=en |volume=323 |issue=1 |pages=1–16 |doi=10.1007/s10509-009-0053-3 |issn=0004-640X|doi-access=free |arxiv=0906.2827 |bibcode=2009Ap&SS.323....1W }}</ref> Patches of cold interstellar clouds with less than 200 AU may have large densities of neutral hydrogen up to 3000&nbsp;cm<sup>−3</sup>, which would not respond to a magnetic field.<ref>{{Cite journal |last1=Meyer |first1=David M. |last2=Lauroesch |first2=J. T. |last3=Peek |first3=J. E. G. |last4=Heiles |first4=Carl |date=2012-06-20 |title=The Remarkable High Pressure of the Local Leo Cold Cloud |arxiv=1204.5980v1 |journal=The Astrophysical Journal |volume=752 |issue=2 |pages=119 (15pp) |doi=10.1088/0004-637X/752/2/119 |doi-access=free|bibcode=2012ApJ...752..119M }}</ref> For a high speed mission to [[Alpha Centauri]] with initial velocity before deceleration <math>v_0=c/10</math> using a coil mass of 1500 tons and a coil radius of <math>R</math>=1600&nbsp;km, the estimated stopping distance <math>x_{max}</math> was 0.37 light years and the total travel time of 58 years with 1/3 of that being deceleration.

In 2017, Crowl documented a design for a mission starting near the Sun and destined for [[Planet Nine|Planet nine]] approximately 1,000 AU distant<ref name=":29" /> that employed the Magsail kinematic model. The design accounted for the Sun's gravity as well as the impact of elevated temperature on the superconducting coil, composed of meta-stable [[metallic hydrogen]], which has a mass density of 3,500&nbsp;kg/m<sup>3</sup> about half that of other superconductors. The mission profile used the Magsail to accelerate away from 0.25 to 1.0 AU from the Sun and then used the Magsail to brake against the Local ISM on approach to Planet nine for a total travel time of 29 years. Parameters and coil mass models differ from those used by Freeland.

Another mission profile uses a magsail oriented at an attack angle to achieve heliocentric transfer between planets moving away from or toward the Sun. In 2013 Quarta and others<ref name=":223">{{Cite journal |last1=Quarta |first1=Alessandro A. |last2=Mengali |first2=Giovanni |last3=Aliasi |first3=Generoso |date=2013-08-01 |title=Optimal control laws for heliocentric transfers with a magnetic sail |url=https://www.sciencedirect.com/science/article/pii/S0094576513001306 |journal=Acta Astronautica |language=en |volume=89 |pages=216–225 |bibcode=2013AcAau..89..216Q |doi=10.1016/j.actaastro.2013.04.018 |issn=0094-5765 |hdl=11568/208940|hdl-access=free }}</ref> used Kajimura 2012 simulation results<ref name=":31" /> that described the lift (steering angle) and torque to achieve a Venus to Earth transfer orbit of 380 days with a coil radius of <math>R_c</math>~1&nbsp;km with characteristic acceleration <math>a_c</math>=1&nbsp;mm/s<sup>2</sup>. In 2019 Bassetto and others<ref name=":322">{{Cite journal |last1=Bassetto |first1=Marco |last2=Quarta |first2=Alessandro A. |last3=Mengali |first3=Giovanni |date=2019-09-01 |title=Magnetic sail-based displaced non-Keplerian orbits |url=https://www.sciencedirect.com/science/article/pii/S1270963819310314 |journal=Aerospace Science and Technology |language=en |volume=92 |pages=363–372 |doi=10.1016/j.ast.2019.06.018 |bibcode=2019AeST...92..363B |issn=1270-9638 |s2cid=197448552 |hdl=11568/1008152|hdl-access=free }}</ref> used the Quarta "thick" magnetopause model and predicted a Venus to Earth transfer orbit of approximately 8 years for a coil radius of <math>R_c</math>~1&nbsp;km. with characteristic acceleration <math>a_c</math>=0.1&nbsp;mm/s<sup>2</sup>. In 2020 Perakis<ref name=":332"/> used the [[#Magsail kinematic model (MKM)|Magsail kinematic model]] with a coil radius of <math>R_c</math>=350 m, current <math>I_c</math>=10<sup>4</sup> A and spacecraft mass of 600&nbsp;kg that changed attack angle to accelerate away from the Earth orbit and decelerate to Jupiter orbit within 20 years.

=== Mini-magnetospheric plasma propulsion (M2P2) ===
[[File:Winglee M2P2 schematic.jpg|thumb|upright=1.5|Winglee M2P2 schematic]]
In 2000, Winglee, Slough and others proposed a design order to reduce the size and weight of a magnetic sail well below that of the [[#Magsail (MS)|magsail]] and named it mini-magnetospheric plasma propulsion (M2P2) that reported results adapted from a simulation model of the Earth's magnetosphere.<ref name=":2" /> The calculates speeds of 50 to 80&nbsp;km s−1 could enable spacecraft:<ref>Mini‐Magnetospheric Plasma Propulsion: Tapping the energy of the solar wind for spacecraft propulsion - Winglee - 2000 - [[Journal of Geophysical Research: Space Physics]]- Wiley Online Library</ref>

* To travel out of the solar system  
* To travel between the planets for low power requirements of ∼1&nbsp;kW per 100&nbsp;kg of payload and ∼0.5&nbsp;kg fuel consumption per day for acceleration periods of several days to a few weeks.

The figure based upon Winglee,<ref name=":2" /> Hajiwara,<ref name=":16">{{Cite web |last1=Hagiwara |first1=T. |last2=Kajimura |first2=Y. |last3=Oshio |first3=Y. |last4=Funaki |first4=I. |date=July 4–10, 2015 |title=Thrust Measurement of Magneto Plasma Sail with Magnetic Nozzle by Using Thermal Plasma Injection |url=http://electricrocket.org/IEPC/IEPC-2015-461p_ISTS-2015-b-461p.pdf |access-date=June 13, 2022 |website=}}</ref> Arita,<ref name=":17">{{Cite journal |last1=Arita |first1=H. |last2=Nishida |first2=H. |last3=Funaki |first3=I. |date=2014 |title=Magnetohydrodynamic Analysis of Thrust Characteristics on Magneto-Plasma Sail with Plasma Magnetic Field Inflation by Low-Beta Plasma |url=https://www.jstage.jst.go.jp/article/tastj/12/ists29/12_Pb_39/_pdf |journal=Trans. JSASS Aerospace Tech. Japan |volume=12 |issue=ists29 |pages=39–44 |doi=10.2322/tastj.12.Pb_39 |bibcode=2014JSAST..12.Pb39A |via=stage.jst.go.jp|doi-access=free }}</ref> and Funaki<ref name=":222" /> illustrates the M2P2 design, which was the basis of the subsequent [[#Magnetoplasma sail (MPS)|Magneto plasma sail (MPS)]] design. Starting at the center with a [[solenoid]] coil of radius <math>R_H</math> of <math display="inline">N_t</math>=1,000 turns carrying a radio frequency current that generates a [[Helicon (physics)|helicon]]<ref>{{Cite journal |last=Chen |first=Francis F. |date=May 1996 |title=Physics of helicon discharges |url=http://aip.scitation.org/doi/10.1063/1.871697 |journal=Physics of Plasmas |language=en |volume=3 |issue=5 |pages=1783–1793 |doi=10.1063/1.871697 |bibcode=1996PhPl....3.1783C |issn=1070-664X}}</ref> wave that injects plasma fed from a source into a coil of radius <math display="inline">R_c</math> that carries a current of <math>I_c</math>, which generates a magnetic field. The excited injected plasma enhances the magnetic field and generates a miniature magnetosphere around the spacecraft, analogous to the [[Heliosphere#Heliopause|heliopause]] where the Sun injected plasma encounters the interstellar medium, [[coronal mass ejection]]s or the Earth's [[Magnetosphere#Magnetotail|magnetotail]]. The injected plasma created an environment that analysis and simulations showed had a magnetic field with a falloff rate of <math>1/r</math> as compared with the classical model of a <math display="inline">1/r^3</math> falloff rate, making the much smaller coil significantly more effective based upon analysis<ref name=":27">{{Cite web |last=Slough |first=John |date=2001 |title=High Beta Plasma for Inflation of a Dipolar Magnetic Field as a Magnetic Sail |url=https://earthweb.ess.washington.edu/space/M2P2/iepc.slough.PDF |archive-url=https://ghostarchive.org/archive/20221009/https://earthweb.ess.washington.edu/space/M2P2/iepc.slough.PDF |archive-date=2022-10-09 |access-date=July 3, 2022 |website=earthweb.ess.washington.edu}}</ref> and simulation.<ref name=":2" /> The pressure of the inflated [[Plasma (physics)|plasma]] along with the stronger magnetic field pressure at a larger distance due to the lower falloff rate would stretch the magnetic field and more efficiently inflate a magnetospheric bubble around the spacecraft.

Parameters for the coil and solenoid were <math>R_H</math>=2.5&nbsp;cm and for the coil <math display="inline">R_c</math>= 0.1 m, 6 orders of magnitude less than the magsail coil with correspondingly much lower mass. An estimate for the weight of the coil was 10&nbsp;kg and 40&nbsp;kg for the plasma injection source and other infrastructure. Reported results from Figure 2 were <math>B_0 \approx</math> {{e|−3}} T at <math>R_{mp} \approx</math> 10&nbsp;km and from Figure 3 an extrapolated result with a plasma injection jet force <math display="inline">F_{jet} \approx</math>10<sup>−3</sup> N resulting in a thrust force of <math display="inline">F_{M2P2} \approx </math>  1 N. The magnetic-only sail force from equation {{EquationNote|MHD.3}} is <math>F_w</math>=3x10<sup>−11</sup> N and thus M2P2 reported a thrust gain of 4x10<sup>10</sup> as compared with a magnetic field only design. Since M2P2 injects ionized gas at a <math>m_{in}</math> [[Mass flow rate|mass flow rate (kg/s)]] it is viewed as a propellant and therefore has a ''[[specific impulse|specific impulse (s)]]'' <math>I_{sp}=F_{M2P2}/m_{in}/g_0</math> where <math>g_0</math> is the acceleration of [[Earth's gravity]].  Winglee stated <math>m_{in}</math>=0.5&nbsp;kg/day and therefore <math>I_{sp}</math>=17,621. The equivalent exhaust velocity <math>v_e=g_0 \, I_{sp}</math> is 173&nbsp;km/s for 1 N of thrust force. Winglee assumed total propellant mass of 30&nbsp;kg and therefore propellant would run out in 60 days.<ref name=":2" />

In 2003, Khazanov published [[Magnetohydrodynamics|MagnetoHydroDynamic]] (MHD) and kinetic studies<ref name=":9" /> that confirmed some aspects of M2P2 but raised issues that the sail size was too small, and that very small thrust would result and also concluded that the hypothesized <math display="inline">1/r</math> magnetic field falloff rate was closer to <math display="inline">1/r^2</math>. The plasma density plots from Khazanov indicated a relatively high density inside the magnetospheric bubble as compared with the external solar wind region that differed significantly from those published by Winglee where the density inside the magnetospheric bubble was much less than outside in the external solar wind region.

A detailed analysis by Toivanen and others in 2004<ref name=":8" /> compared a theoretical model of Magsail, dubbed Plasma-free Magnetospheric Propulsion (PFMP) versus M2P2 and concluded that the thrust force predicted by Winglee and others was over ten orders of magnitude optimistic since the majority of the solar wind momentum was delivered to the magnetotail and current leakages through the magnetopause and not to the spacecraft.<ref name=":15">{{Cite web |last=Janhunen |first=P. |date=October 11, 2002 |title=Comment on: "Mini-magnetospheric plasma propulsion: tapping the energy of the solar wind for spacecraft propulsion" by R. Winglee et al. |url=https://space.fmi.fi/~pjanhune/papers/winglee.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://space.fmi.fi/~pjanhune/papers/winglee.pdf |archive-date=2022-10-09 |access-date=June 28, 2022 |website=space.fmi.fi}}</ref> Their comments also indicated that the magnetic field lines may not close near enough to the coil to achieve significant transfer of force. Their analysis made an analogy to the [[Heliospheric current sheet]] as an example in astrophysics where the magnetic field could falloff at a rate of between <math display="inline">1/r</math> and <math display="inline">1/r^2</math>. They also analyzed current sheets reported by Winglee from the magnetopause to the spacecraft in the windward direction and a current sheet in the magnetotail. Their analysis indicated that the current sheets needed to pass extremely close to the spacecraft to impart significant force could generate significant heat and render this leverage impractical.

In 2005, Cattell and others<ref name="05ja026_full" /> published comments regarding M2P2 that included a lack of magnetic flux conservation in the region outside the magnetosphere that was not considered in the Khazanov studies. Their analysis concluded in Table 1 that Winglee had significantly underestimated the required sail size, mass, and required magnetic flux. Their analysis asserted that the hypothesized <math display="inline">1/r</math> magnetic field falloff rate was not possible.

The expansion of the magnetic field using injected plasma was demonstrated in a large vacuum chamber on [[Earth]], but quantification of thrust was not part of the experiment.<ref>{{Cite web |last=Winglee |first=R.M. |date=November 2001 |title=Mini-Magnetospheric Plasma Propulsion (M2P2) NIAC Award No. 07600-032: Final Report |url=http://www.niac.usra.edu/files/studies/final_report/372Winglee.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.niac.usra.edu/files/studies/final_report/372Winglee.pdf |archive-date=2022-10-09 |access-date=July 7, 2022 |website=niece.usra.edu}}</ref> The accompanying presentation has some good animations that illustrate physical principles described in the report.<ref>{{Cite web |last1=Winglee |first1=R.M. |last2=Ziemba |first2=T. |last3=Slough |first3=J. |last4=Euripedes |first4=P. |date=June 2001 |title=Mini-Magnetic Plasma Propulsion [M2P2] |url=http://www.niac.usra.edu/files/library/meetings/annual/jun01/372Winglee.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.niac.usra.edu/files/library/meetings/annual/jun01/372Winglee.pdf |archive-date=2022-10-09 |access-date=July 7, 2022 |website=niece.usr.edu}}</ref>  A 2004 Winglee paper focused on usage of M2P2 for electromagnetic shielding.<ref>{{Cite journal |last=Winglee |first=Robert |date=2004 |title=Advances in Magnetized Plasma Propulsion and Radiation Shielding |journal=Proceedings of the 2004 NASA.DoD Conference on Evolution Hardware |citeseerx=10.1.1.513.2375 }}</ref> Beginning in 2003, the [[#Magnetoplasma sail (MPS)|Magneto plasma sail]] design further investigated the plasma injection augmentation of the magnetic field, used larger coils<ref name=":20" /> and reported significantly more modest gains.

=== 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" />

=== Plasma magnet (PM) ===
[[File:Plasma magnet principles of operation.jpg|thumb|upright=1.5|Plasma magnet principles of operation]]
The plasma magnet (PM) sail design introduced a different approach to generate a static magnetic dipole as illustrated in the figure.<ref name=":18" /><ref name="Slough2006" /> As shown in the detailed view on the right the field source is two relatively small crossed perpendicularly oriented antenna coils each of radius <math>R_c</math> (m), each carrying a sinusoidal [[alternating current]] (AC) with the total current of ''<math>I_c</math>'' (A) generated by an onboard power supply. The AC current applied to each coil is out of phase by 90° and consequently generates a [[rotating magnetic field]] (RMF) with [[rotational speed|rotational frequency (s<sup>−1</sup>)]] <math>\omega_{RMF}</math> chosen that is fast enough that positive ions do not rotate but the less massive electrons rotate at this speed. The figure illustrates rotation using color coded contours of constant magnetic strength, not magnetic field lines. In order to inflate the magnetospheric bubble the thermal [[plasma beta]] <math>\beta_t</math> must be high and initially a plasma injection may be necessary, analogous to inflating a balloon when small and internal tension is high. After initial inflation, protons and rotating electrons are captured from the plasma wind through the leaky magnetopause and as shown in the left create a current disc shown as transparent red in the figure with darker shading indicating greatest density near the coil pair and extending out to the magnetopause radius ''R<sub>mp</sub>'', which is orders of magnitude larger than the coil radius ''R<sub>c</sub>'' (figure not drawn to scale). See RMDCartoon.avi for an animation of this effect.<ref>{{Cite web |date=Nov 2011 |title=Plasma Magnet |url=https://earthweb.ess.washington.edu/space/PlasmaMag/ |access-date=July 17, 2022 |website=earthweb.ess.washington.edu}}</ref> The induced current disc carries a direct current <math>I_{cd}</math> orders of magnitude larger than the input alternating current ''<math>I_c</math>'' and forms a static dipole magnetic field oriented perpendicular to the current disc reaching a standoff balance with the plasma wind pressure at distance <math>R_{mp} \approx L</math> at the magnetopause boundary according to the [[#Magnetohydrodynamic model|MHD model of an artificial magnetosphere]].

The magnetic field falloff rate was assumed in 2001<ref name="Slough2006" />{{Rp|location=Eq (7)}} and 2006<ref name=":27" />{{Rp|location=Eq (8)}} to be <math>f_o </math> =1. However, as described by Khazanov in 2003<ref name=":9" /> and restated by Slough, Kirtley and Pancotti in 2011<ref name=":22">{{Cite journal |last1=Slough |first1=J. |last2=Kirtley |first2=D. |last3=Pancotti |first3=A. |last4=Llc |first4=Msnw |date=2011 |title=Plasma Magnetoshell for Aerobraking and Aerocapture |journal=Iepc-2011-304 |s2cid=99132947}}</ref>{{Rp|location=Eq (2)}} and Kirtley and Slough in a 2012 NIAC report<ref name=":7" />{{Rp|location=Eq (4)}}  that <math>f_o </math>=2 as demanded by [[Alfvén's theorem#Flux conservation|conservation of magnetic flux]]. Several [[#Magnetoplasma sail (MPS)|MPS]] studies concluded that <math>f_o </math> is closer to 2. The falloff rate <math>f_o </math> is a critical parameter in the determination of performance.

The RMF-induced rotating disc of electrons has [[Current density|current density (A m<sup>−2</sup>)]] <math>j_\theta(r)</math> at distance ''r'' from the antenna for <math>f_o=1</math><ref name="Slough2006" />{{Rp|location=Eq (5)}}  and for <math>f_o=2</math>,<ref name=":22" />{{Rp|location=Eq (6)}} which states that flux conservation requires this falloff rate, consistent with a criticism of M2P2 by Cattell<ref name="05ja026_full" /> as follows:
{{NumBlk2|:|<math>j_\theta(r,f_o)=\frac{2 f_o \, B(R_0) \, R_0^{f_o} }{\mu_0 \, r^{f_o+1} } \, \, ,r>R_0</math>|PM.1}}
where <math display="inline">B(R_0) </math> T is the magnetic field flux density at radius <math>R_0 \approx R_c </math> m near the antenna coils. Note that the current density is highest at <math>r=R_0</math> and falls off at a rate of <math>f_o+1</math>. A critical condition for the plasma magnet design<ref name="Slough2006" />{{Rp|location=Eq (1a)}} provides a lower bound on the RMF frequency <math>\omega_{RMF} </math> rad/s as follows so that electrons in the plasma wind are magnetized and rotate but the ions are not magnetized and do not rotate:
{{NumBlk2|:|<math>\omega_{RMF}>\omega_{ci} = \frac {Z \, e \, B(R_0)}{m_i}</math>|PM.2}}
where <math>\omega_{ci} </math> is the [[Ion cyclotron resonance#Definition of the resonant frequency|ion gyrofrequency (s<sup>−1</sup>)]] in the RMF near the antenna coils, <math>Z </math> is charge number of the ion, <math>e </math> is the [[elementary charge]], and <math>m_i </math> kg is the (average) mass of the ion(s). Specifying the magnetic field near the coils at radius <math>R_0</math> is critical since this is where the current density is greatest. Choosing a magnetic field at magnetopause yields a lower value of <math>\omega_{RMF} </math> but ions closer to the coils will rotate. Another condition is that <math>\omega_{RMF} </math> be small enough such that collisions are extremely unlikely.

The required power to generate the RMF  <math>P_{RMF}</math> is derived by integrating the product of the square of the current density from equation {{EquationNote|PM.1}} and the resistivity of the plasma <math>\eta_p</math> from <math>R_0</math> to <math>R_{mp}</math> with the result as follows:
{{NumBlk2|:|<math>P_{RMF}\approx \frac {4 \, \pi \, \eta_p}{\mu_0^2} \, B(R_0)^2 \, R_0 \,  \frac{f_o^2}{2f_o-1}</math>|PM.3}}
where <math>\eta_p</math> is the [[Spitzer resistivity|Spitzer resistivity (W m)]]<ref name=":33">{{Cite web |last=Richardson |first=A. S. |date=2019 |title=2019 NRL Plasma Formulary |url=https://www.nrl.navy.mil/Portals/38/PDF%20Files/NRL_Formulary_2019.pdf?ver=p9F4Uq9wAtB0MPBwKYL9lw== |access-date=July 26, 2022 |website=nrl.navy.mil}}</ref><!-- DM: Confirm correct units for Spitzer resistivity. --> of the plasma of ~1.2x10<sup>−3</sup> <math>T_e^{-3/2}</math> where ''<math>T_e</math>'' is the electron temperature assumed to be 15 eV,<ref name="Slough2006" /> the same result for <math>f_o=1</math><ref name="Slough2006" />{{Rp|location=Eq (7)}} and for <math>f_o=2</math>.<ref name=":22" />{{Rp|location=Eq (7)}}

Starting with the definition of plasma wind force from equation {{EquationNote|MFM.5}}, noting that <math>P_w=F_w u</math> rearranging and recognizing that equation {{EquationNote|PM.3}} gives the solution for <math>B(R_0)</math>, which can be substituted and then using equation {{EquationNote|MHD.2}} for <math>B_{mp}</math> yields the following expression
{{NumBlk2|:|<math>P_w = \frac{C_d}{2\,  C_{SO} } \rho \, u^3 \, \pi \, R_0^2 \,
\biggl(\frac {P_{RMF} }{u^2\, R_0 \, \rho \, \mu_0} \biggr)^{1/f_o}
\biggl(\frac {\mu_0^2}{4 \pi \eta_p} \frac {2f_o-1}{f_o^2} \biggr)^{1/f_o}</math>|PM.4}}
which when multiplied by <math>C_d/2</math> with <math>C_SO=1</math> is the same as for <math>f_o=1</math><ref name="Slough2006" />{{Rp|location=Eq (10)}} Note that solution for <math>P_{RMF}</math> and <math>R_0</math> must also satisfy equation {{EquationNote|MHD.3}}, to which the comments following<ref name="Slough2006" />{{Rp|location=Eq (10)}} regarding a "tremendous leverage of power" do not address.

Note that a number of the examples cited in<ref name="Slough2006" /> assume a magnetopause radius <math>R_{mp}</math> that do not meet the MHD applicability test of equation {{EquationNote|MHD.5}}.  From the [[Power (physics)#Definition|definition of power in physics]] a constant force is power divided by velocity, the force generated by the plasma magnet (PM) sail is as follows from equation  {{EquationNote|PM.4}}{{NumBlk2|:|<math>F_{PM} =  \frac {P_w}{u} =  \frac{C_d}{2\,  C_{SO} } \rho \, u^2 \, \pi \, R_0^2 \,
\biggl(\frac {P_{RMF} }{ u^2 \, R_0 \, \rho \, \mu_0} \biggr)^{1/f_o} 
\biggl(\frac {\mu_0^2}{4 \pi \eta_p} \frac {2f_o-1}{f_o^2} \biggr)^{1/f_o}</math>|PM.5}}Comparing the above with Equation {{EquationRef|(MFM.6)}} not the dependence on plasma mass density <math>\rho</math> is of the same form <math>\rho ^{1-1/f_o}</math>. Note from Equation {{EquationNote|PM.5}} that as the falloff rate <math>f_o</math> increases that the force derived from the plasma wind decreases, or to maintain the same force <math>P_{RMF}</math> and/or <math>R_0</math> must increase to maintain the same force <math>F_{PM}</math>.

Equation {{EquationNote|CMC.2}} gives the mass for each physical coil of radius <math>R_c</math> m. Since the RMF requires alternating current and semiconductors are not efficient at higher frequencies, aluminum was specified with mass density <math>\delta_c</math> = 2,700&nbsp;kg/m<sup>3</sup>. Estimates of the coil mass<ref name="Slough2006" /> are optimistic by a factor of <math>4 \pi</math> since only one coil was sized and the coil circumference was specified as <math>R_c</math> instead of <math>2 \pi R_c</math>.

The coil [[Electrical resistivity and conductivity#Superconductivity|resistance]] is the product of coil material [[Electrical resistivity and conductivity|resistivity (Ω m)]] <math>\eta_c</math>  (e.g., ~3x10<sup>−8</sup> Ωm for aluminum) and the coil length <math>2 \pi R_c</math> divided by the coil wire cross sectional area where <math>r_c</math> is the radius of the coil wire as follows:
{{NumBlk2|:|<math>\Omega_c = \frac{\eta_c 2 \pi R_c}{\pi r_c^2} = \frac {2 \eta_c R_c}{r_c^2}</math>|PM.6}}
Some additional power must compensate for resistive loss but it is orders of magnitude less than <math>P_{RMF}</math>. The peak current carried by a coil is specified by the RMF power and coil resistance from the definition of [[Power (physics)#Electrical power|electrical power in physics]] as follows:
{{NumBlk2|:|<math>I_c = \sqrt{\frac {P_{RMF} }{\Omega_c} }</math>|PM.7}}
The current induced in the disc by the RMF <math>I_{ic}</math> is the integral of the current density <math>j(r,f_o)</math> from equation {{EquationNote|PM.1}} on the surface of the disc with inner radius <math display="inline">R_c</math> and outer radius <math display="inline">R_{MP}</math> with result:{{NumBlk2|:|<math>I_{ic} = \frac {2 f_o \pi \, B(R_c)}{\mu_0} \begin{cases} R_c \ln(R_{mp}/R_0) &\text{, }f_o{=1} \\ 1-R_c/R_{mp} & \text{, }f_o{=2}\end{cases}</math>|PM.8}}
the same result for <math>f_o</math>=1.<ref name="Slough2006" />{{Rp|location=Eq 11}}<!-- DM: Same result for f_o=2 in Kirtley 2012? -->

Laboratory experiments<ref name="Slough2006" /> validated that the RMF creates a magnetospheric bubble, electron temperature near the coils increases indicating presence of the rotatting disc of electrons and that thrust was generated. Since the scale of a terrestrial experiment is limited, simulations or a flight trial was recommended. Some of these concepts adapted to an ionospheric plasma environment were carried on in the [[#Plasma magnetoshell (PMS)|plasma magnetoshell]] design.

In 2022 Freeze, Greason and others<ref name=":11" /> published a detailed design for a [[plasma magnet]] based sail for a spacecraft named Wind Rider that would use solar wind force to accelerate away from near Earth and decelerate against the magnetosphere of Jupiter in a spaceflight trial mission called Jupiter Observing Velocity Experiment (JOVE). This design employed a pair of superconducting coils each with radius <math>R_c</math> of 9 m, an alternating current of <math>I_c</math> of 112 A with <math>\omega_{RMF}/(2\pi)</math> and a falloff rate of <math>f_o=1</math>.<ref name=":11" />{{Rp|location=Eq (5)}} A transit time to Jupiter of 25 days was reported for a 21&nbsp;kg spacecraft design launched in a 16 U Cubesat format.

Using <math>f_o</math>=1 creates very optimistic performance numbers, but since Slough changed this to <math>f_o</math>=2 in 2011<ref name=":22" /> and 2012,<ref name=":7" />  the case of <math>f_o=1</math> is not compared in this article. An example for <math>f_o</math>=2 using solar wind parameters <math>\rho</math>=8x10<sup>−21</sup> kg/m<sup>3</sup>, <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>R_{mp}</math>=10<sup>5</sup> m results in <math>r_g < L</math> where MHD applicability occurs.  With a coil radius of <math>R_c</math>=1,000 m yields <math>B_0</math>=4x10<sup>−4</sup> T from equation {{EquationNote|MFM.2}}. The required RMF power from equation {{EquationNote|PM.3}} is 13&nbsp;kW with a required AC coil current <math>I_c</math>=10 A from equation {{EquationNote|PM.3}} resulting in an induced current of <math>I_{ic}</math>=2 kA from equation {{EquationNote|PM.7}} . With <math>C_d</math>=5 the plasma magnet force from equation {{EquationNote|PM.3}}  is 197 N. The magnetic force only for the above parameters is 2.8 N from equation {{EquationNote|MFM.5}} and therefore the plasma magnet thrust gain is 71. The [[#Performance comparison|performance comparison]] section gives and optimistic estimate using constant acceleration for  <math>f_o</math>=2 results in a transit time of ~100 days.<!-- DM: Confirm that parameter choices in the para match PM in comparison table. -->

=== Plasma magnetoshell (PMS) ===
A 2011 paper by Slough and others<ref name=":22" /> and a 2012 NIAC report by Kirtley<ref name=":7" /> investigated use of the plasma magnet technology with 1/r<sup>2</sup> magnetic field falloff rate for use in the ionosphere of a planet as a braking mechanism in an approach dubbed Plasma magnetoshell (PMS). The magnetoshell creates drag by ionizing neutral atoms in a planet's ionosphere then magnetically deflecting them. A tether attaching the plasma magnet coils to the spacecraft transfers momentum such that orbital insertion occurs. Analytical models, laboratory demonstrations and mission profiles to Neptune and Mars were described.

In 2017, Kelly described using a single-coil magnet with 1/r<sup>3</sup> magnetic field falloff rate and more experimental results.<ref>{{Cite journal |last1=Kelly |first1=C |last2=Shimazu |first2=Akihisa |date=October 2017 |title=Revolutionizing Orbit Insertion with Active Magnetoshell Aerocapture |url=http://electricrocket.org/IEPC/IEPC_2017_600.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://electricrocket.org/IEPC/IEPC_2017_600.pdf |archive-date=2022-10-09 |journal=35th International Electric Propulsion Conference |location=Atlanta, GA}}</ref> In 2019 Kelly and Little published simulation results for magnetoshell performance scaling.<ref name=":35">{{Cite book |last1=Kelly |first1=Charles L. |last2=Little |first2=Justin M. |title=2019 IEEE Aerospace Conference |chapter=Energy and Mass Utilization During Drag-Modulated Plasma Aerocapture |date=March 2019 |chapter-url=https://ieeexplore.ieee.org/document/8741698 |location=Big Sky, Montana|publisher=IEEE |pages=1–10 |doi=10.1109/AERO.2019.8741698 |isbn=978-1-5386-6854-2|s2cid=195225221 }}</ref>  A magnet with radius <math>R_c</math>=1 m was tethered to a spacecraft with batteries for 1,000 seconds of operation (longer than aerocapture designs). The simulations assumed a magnet mass <math>M_c</math>=1,000&nbsp;kg and total magnetoshell system mass of 1,623&nbsp;kg, suitable for a [[Cassini–Huygens]] or [[Juno (spacecraft)|Juno]] size orbiter. The planet's mass and atmosphere atomic composition and density determine a threshold velocity where magnetoshell operation is feasible. Saturn and Neptune have a hydrogen atmosphere and a threshold velocity of approximately 22&nbsp;km/s. In a Neptune mission a <math>\Delta v</math>=6&nbsp;km is required for a 5,000&nbsp;kg spacecraft and must average 50&nbsp;kN for the maneuver duration. The model overestimates performance for N<sub>2</sub> (Earth, Titan) and CO<sub>2</sub> (Venus, Mars) atmospheres since multiple ion species are created and more complex interactions occur. Furthermore, the relatively lower mass of Venus and Mars reduces the threshold velocity below that of feasible magnetoshell operation. The paper states that [[aerocapture]] technologies are mature enough for these mission profiles.

In 2021, Kelly and Little published further details<ref name=":36">{{Cite book |last1=Kelly |first1=Charles L. |last2=Little |first2=Justin M. |title=2021 IEEE Aerospace Conference (50100) |chapter=Performance and Design Scaling of Magnetoshells for Outer Planet Drag-Modulated Plasma Aerocapture |date=2021-03-06 |chapter-url=https://ieeexplore.ieee.org/document/9438387 |location=Big Sky, Montana |publisher=IEEE |pages=1–10 |doi=10.1109/AERO50100.2021.9438387 |isbn=978-1-7281-7436-5|s2cid=235383575 }}</ref> for use of drag-modulated plasma aerocapture (DMPA) that when compared to Adaptable Deployable Entry and Placement Technology (ADEPT)<ref>{{Cite web |last=Wercinski |first=P. |date=April 23, 2019 |title=A Neptune Orbiter Concept Using Drag Modulated Aerocaptue (DMA) And The Adaptable, Deployable Entry And Placement Technology (ADEPT) |url=https://ntrs.nasa.gov/citations/20190029585 |access-date=September 16, 2022 |website=ntrs.nasa.gov}}</ref> for drag-modulated aerocapture (DMA) to Neptune<ref>{{Cite web |last=Venkatapathy |first=E |date=January 22, 2020 |title=Enabling Entry Technologies For Ice Giant Missions |url=https://ntrs.nasa.gov/citations/20200000499 |access-date=September 16, 2022 |website=ntrs.nasa.gov}}</ref> that could deliver 70% higher orbiter mass and experience 30% lower stagnation heating.

=== Beam powered magsail (BPM) ===
A [[Beam-powered propulsion|beam-powered]] of M2P2 variant, [[Magnetized beamed plasma propulsion|MagBeam]] was proposed in 2011.<ref>{{Cite web |title=MagBeam |url=https://earthweb.ess.washington.edu/space/magbeam/ |website=earthweb.ess.washington.edu}}</ref> In this design a magnetic sail is used with [[beam-powered propulsion]], by using a high-power [[particle accelerator]] to fire a beam of charged particles at the spacecraft.<ref>G. Landis, "Interstellar Flight by Particle Beam," ''Acta Astronautica. Vol 55'', No. 11, 931–934 (December 2004).</ref> The magsail would deflect this beam, transferring momentum to the vehicle, that could provide higher acceleration than a solar sail driven by a [[laser]], but a charged particle beam would disperse in a shorter distance than a laser due to the electrostatic repulsion of its component particles.  This dispersion problem could potentially be resolved by accelerating a stream of sails which then in turn transfer their momentum to a magsail vehicle, as proposed by [[Jordin Kare]].{{Citation needed|date=September 2022|reason=Need citation to reliable source.}}