{"history":[],"article":{"id":"b4426579-0703-4e9e-9e19-00970c2ea468","title":"Magnetic sail","slug":"magnetic-sail","content":":::summary\n:::\n\n# \nA **magnetic sail** is a proposed method of [spacecraft propulsion](https://bliptext.com/articles/spacecraft-propulsion) where an onboard magnetic field source interacts with a [plasma](https://bliptext.com/articles/plasma-physics) wind (e.g., the [solar wind](https://bliptext.com/articles/solar-wind)) to form an [artificial magnetosphere](https://bliptext.com/articles/) (similar to Earth's [magnetosphere](https://bliptext.com/articles/magnetosphere)) that acts as a sail, transferring force from the wind to the spacecraft requiring little to no propellant as detailed for each proposed **magnetic sail** design in this article.\nThe animation and the following text summarize the [magnetic sail physical principles](https://bliptext.com/articles/magnetic-sail) involved.\nThe spacecraft's [magnetic field](https://bliptext.com/articles/magnetic-field) source, represented by the purple dot, generates a [magnetic field](https://bliptext.com/articles/magnetic-field), shown as expanding black circles.\nUnder conditions summarized in the [overview](https://bliptext.com/articles/) section, this field creates a magnetosphere whose leading edge is a [magnetopause](https://bliptext.com/articles/magnetopause) and a [bow shock](https://bliptext.com/articles/bow-shock) composed of [charged particles](https://bliptext.com/articles/charged-particle) captured from the wind by the magnetic field, as shown in blue, which deflects subsequent [charged particles](https://bliptext.com/articles/charged-particle) from the plasma wind coming from the left.\nSpecific attributes of the [artificial magnetosphere](https://bliptext.com/articles/) around the spacecraft for a specific design significantly affect performance as summarized in the overview section.\nA [magnetohydrodynamic model](https://bliptext.com/articles/magnetic-sail) (verified by computer simulations and laboratory experiments) predicts that the interaction of the artificial magnetosphere with the oncoming plasma wind creates an effective sail blocking area that transfers force as shown by a sequence of labeled arrows from the plasma wind, to the spacecraft's magnetic field, to the spacecraft's field source, which accelerates the spacecraft in the same direction as the plasma wind.\nThese concepts apply to all [proposed magnetic sail system designs](https://bliptext.com/articles/), with the difference how the design generates the magnetic field and how efficiently the field source creates the [artificial magnetosphere](https://bliptext.com/articles/) described above.\nThe [History of concept](https://bliptext.com/articles/) section summarizes key aspects of the proposed designs and relationships between them as background.\nThe cited references are technical with many equations and in order to make the information more accessible, this article first describes in text (and illustrations where available) beginning in the overview section and prior to each design, section or groups of equations and plots intended for the technically oriented reader.\nThe beginning of each proposed design section also contains a summary of the important aspects so that a reader can skip the equations for that design.\nThe differences in the designs determine [performance measures](https://bliptext.com/articles/), such as the mass of the field source and necessary power, which in turn determine force, mass and hence acceleration and velocity that enable a [performance comparison](https://bliptext.com/articles/) between magnetic sail designs at the end of this article.\nA [comparison with other spacecraft propulsion methods](https://bliptext.com/articles/spacecraft-propulsion) includes some magnetic sail designs where the reader can click on the column headers to compare magnetic sail performance with other propulsion methods.\nThe following observations result from this comparison: magnetic sail designs have insufficient thrust to launch from Earth, thrust (drag) for deceleration for the [magsail](https://bliptext.com/articles/) in the interstellar medium is relatively large, and both the [magsail](https://bliptext.com/articles/) and [magnetoplasma sail](https://bliptext.com/articles/) both have significant thrust for travel away from Earth using the force from the solar wind.\n\n# History of concept\nAn overview of many of the magnetic sail proposed designs with illustrations from the references was published in 2018 by Djojodihardjo.\nThe earliest method proposed by Andrews and Zubrin in 1988, dubbed the [magsail](https://bliptext.com/articles/), has the significant advantage of requiring no propellant and is thus a form of [field propulsion](https://bliptext.com/articles/field-propulsion) that can operate indefinitely.\nA drawback of the magsail design was that it required a large (50–100 km radius) superconducting loop carrying large currents with a mass on the order of 100 tonnes.\nThe magsail design also described [modes of operation](https://bliptext.com/articles/) for interplanetary transfers, thrusting against a planetary [ionosphere](https://bliptext.com/articles/ionosphere) or [magnetosphere](https://bliptext.com/articles/magnetosphere), escape from low Earth orbit as well as deceleration of an interstellar craft over decades after being initially accelerated by other means, for example.\na [fusion rocket](https://bliptext.com/articles/fusion-rocket), to a significant fraction of light speed, with a more detailed design published in 2000.\nIn 2015, Freeland validated most of the initial magsail analysis, but determined that thrust predictions were optimistic by a factor of 3.1 due to a numerical integration error.\nSubsequent designs proposed and analyzed means to significantly reduce mass.\nThese designs require little to modest amounts of exhausted propellant and can thrust for years.\nAll proposed designs describe thrust from solar wind outwards from the Sun.\nIn 2000, Winglee and Slough proposed a [Mini-Magnetospheric Plasma Propulsion (M2P2)](https://bliptext.com/articles/) design that injected low energy plasma into a much smaller coil with much lower mass that required low power.\nSimulations predicted impressive performance relative to mass and required power; however, a number of critiques raised issues: that the assumed magnetic field falloff rate was optimistic and that thrust was dramatically overestimated.\nStarting in 2003, Funaki and others published a series of theoretical, simulation and experimental investigations at JAXA in collaboration with Japanese universities addressing some of the issues from criticisms of M2P2 and named their approach the [MagnetoPlasma Sail (MPS)](https://bliptext.com/articles/).\nIn 2011, Funaki and Yamakawa authored a chapter in a book that is a good reference for magnetic sail theory and concepts.\nMPS research resulted in many published papers that advanced the understanding of [physical principles for magnetic sails](https://bliptext.com/articles/).\nBest performance occurred when the injected plasma had a lower density and velocity than considered in M2P2.\nThrust gain was computed as compared with performance with a magnetic field only in 2013 and 2014.\nInvestigations and experiments continued reporting increased thrust experimentally and numerically considering use of a [Magnetoplasmadynamic thruster](https://bliptext.com/articles/magnetoplasmadynamic-thruster) (aka MPD Arc jet in Japan)in 2015, multiple antenna coils in 2019, and a multi-pole MPD thruster in 2020.\nSlough published in 2004 and 2006 a method to generate the static magnetic dipole for a magnetic sail in a design called the [Plasma magnet (PM)](https://bliptext.com/articles/) that was described as an [AC induction motor](https://bliptext.com/articles/induction-motor) turned inside out.\nA pair of small perpendicularly oriented coils acted as the stator powered by an alternating current to generate a [rotating magnetic field (RMF)](https://bliptext.com/articles/rotating-magnetic-field) that analysis predicted and laboratory experiments demonstrated that a current disc formed as the rotor outside the stator.\nThe current disk formed from electrons captured from the plasma wind, therefore requiring little to no plasma injection.\nPredictions of substantial improvements in terms of reduced coil size (and hence mass) and markedly lower power requirements for significant thrust hypothesized the same optimistic magnetic field falloff rate as assumed for M2P2.\nIn 2022, a spaceflight trial dubbed Jupiter Observing Velocity Experiment (JOVE) proposed using a [plasma magnet](https://bliptext.com/articles/plasma-magnet) based sail for a spacecraft named Wind Rider using the solar wind to accelerate away from a point near Earth and decelerate against the magnetosphere of Jupiter.\nA 2012, study by Kirtley and Slough investigated using the plasma magnet technology to use plasma in a planetary ionosphere as a braking mechanism and was called the Plasma Magnetoshell.\nThis paper restated the magnetic field falloff rate to the value suggested in the critiques of M2P2 that dramatically reduces analytical predicted performance.\nInitial missions targeted deceleration in the ionosphere of Mars.\nKelly and Little in 2019 published simulation results using a multi-turn coil and not the plasma magnet showed that the magnetoshell was viable for orbital insertion asy Mars, Jupiter, Neptune and Uranus and in 2021 showed that it was more efficient than [aerocapture](https://bliptext.com/articles/aerocapture) for Neptune.\nIn 2021, Zhenyu Yang and others published an analysis, numerical calculations and experimental verification for a propulsion system that was a combination of the magnetic sail and the [electric sail](https://bliptext.com/articles/electric-sail) called an electromagnetic sail.\nA superconducting magsail coil augmented by an [electron gun](https://bliptext.com/articles/electron-gun) at the coil's center generates an electric field as in an electric sail that deflects positive ions in the plasma wind thereby providing additional thrust, which could reduce overall system mass.\n\n# Overview\nThe [Modes of operation](https://bliptext.com/articles/) section describes the important parameters of [plasma particle density and wind velocity](https://bliptext.com/articles/) in conjunction with a use case for:\nThe [Physical principles](https://bliptext.com/articles/) section details aspects of how charged particles in a plasma wind interact with a magnetic field and conditions that determine how much thrust force results on the spacecraft in terms of particle's behavior in a plasma wind, as well as the form and magnitude of the magnetic field related to conditions within the magnetosphere that differ for the proposed designs.\n[Charged particles](https://bliptext.com/articles/charged-particle) such as electrons, protons and ions travel in straight lines in a vacuum in the absence of a magnetic field.\nAs shown in the illustration in the presence of a [magnetic field](https://bliptext.com/articles/magnetic-field) shown in green, charged particles gyrate in circular arcs with blue indicating positively charged particles (e.g., protons) and red indicating electrons.\nThe particle's [gyroradius](https://bliptext.com/articles/gyroradius) is proportional to the ratio of the particle's [momentum](https://bliptext.com/articles/momentum) (product of mass and velocity) over the magnetic field.\nAt 1 [Astronomical Unit (AU)](https://bliptext.com/articles/astronomical-unit), the distance from the Sun to the Earth, the gyroradius of a [proton](https://bliptext.com/articles/proton) is ~72 km and since a [proton](https://bliptext.com/articles/proton) is ~1,836 times the mass of an [electron](https://bliptext.com/articles/electron), the gyroradius of an [electron](https://bliptext.com/articles/electron) is ~40 m with the illustration not drawn to scale.\nFor the magsail [deceleration in the interstellar medium (ISM)](https://bliptext.com/articles/) mode of operation the velocity is a significant fraction of light speed, for example 5% c, the gyroradius is ~ 500 km for protons and ~280 m for electrons.\nWhen the magsail magnetopause radius is much less than the proton gyroradius the [magsail kinematic model](https://bliptext.com/articles/) by Gros in 2017, which considered only protons, predicts a marked reduction in thrust force for initial ship velocity greater than 10% c prior to deceleration.\nWhen the magnetosphere radius is much greater than the spacecraft's magnetic field source radius, all proposed designs, except for the magsail, use a [magnetic dipole](https://bliptext.com/articles/magnetic-dipole) approximation for an [Amperian loop](https://bliptext.com/articles/magnetic-moment) shown in the center of the illustration with the X indicating current flowing into the page and the dot indicating current flowing out of the page.\nThe illustration shows the resulting [magnetic field lines](https://bliptext.com/articles/magnetic-field) and their direction, where the closer spacing of lines indicates a stronger field.\nSince the magsail uses a large superconducting coil that has a radius on the same order as the magnetosphere the details of that design use the [magsail MHD model](https://bliptext.com/articles/) employing the [Biot–Savart law](https://bliptext.com/articles/biot-savart-law) that predicts stronger magnetic fields near and inside the coil than the dipole model.\nA [Lorentz force](https://bliptext.com/articles/lorentz-force) occurs only for the portion of a charged particle's velocity at a right angle to the magnetic field lines and this constitutes the magnetic force depicted in the summary animation.\nElectrically neutral particles, such as neutrons, atoms and molecules are unaffected by a magnetic field.\nA condition for [applicability of magnetohydrodynamic (MHD)](https://bliptext.com/articles/) theory, which models charged particles as fluid flows, is that to achieve maximum force the radius of the [artificial magnetosphere](https://bliptext.com/articles/) be on the same order as the ion gyroradius for the plasma environment for a [particular mode of operation](https://bliptext.com/articles/).\nAnother important condition is how the proposed design affects the magnetic field falloff rate inside the magnetosphere, which impacts the field source mass and power requirements.\nFor a radial distance r from the spacecraft's magnetic field source in a vacuum the magnetic field falls off as 1/r^{f_o}, where f_o is the falloff rate.\nClassic [magnetic dipole](https://bliptext.com/articles/magnetic-dipole) theory covers the case of f_o=3 as used in the magsail design.\nWhen plasma is injected and/or captured near the field source, the magnetic field falls off at a rate of, a topic that has been a subject of much research, criticism and differs between designs and has changed over time for the plasma magnet.\nThe M2P2 and plasma magnet designs initially assumed f_o=1 that as shown in numerical examples summarized at the end of the corresponding design sections predicted a very large performance gain.\nSeveral researchers independently created a [magnetic field model](https://bliptext.com/articles/) where and asserted that an f_o=2 falloff rate was the best achievable.\nIn 2011 the [plasma magnet](https://bliptext.com/articles/) author changed the falloff rate f_o from 1 to 2 and that is the value used for the [plasma magnet](https://bliptext.com/articles/) for performance comparison in this article.\nThe [magnetoplasma sail (MPS)](https://bliptext.com/articles/) design is an evolution of the M2P2 concept that has been extensively documented, numerically analyzed and simulated and reported a falloff rate f_o between 1.5 and 2.\nThe falloff rate f_o has a significant impact on performance or the mode of operation [accelerating away from the Sun](https://bliptext.com/articles/) where the mass density of ions in the plasma decreases according to an [Inverse-square law](https://bliptext.com/articles/inverse-square-law) with distance from the Sun (e.g., AU) increases.\nThe illustration shows in a [semi-log plot](https://bliptext.com/articles/semi-log-plot) the impact of falloff rate f_o on relative force F from Equation versus distance from the Sun ranging from 1 to 20 AU, the approximate distance of Neptune.\nThe distance to Jupiter is approximately 5 AU.\nConstant force independent of distance from the Sun for f_o=1 is stated in several plasma magnet references, for example Slough and Freeze and results from the effective increase in sail blocking area to exactly offset reduced plasma mass density as a magnetic sail spacecraft accelerates in response to the plasma wind force away from the Sun.\nAs seen from the illustration the impact of falloff rate f_o on force, and therefore acceleration, becomes grerater as distance from the Sun increases.\nAt scales where the artificial magnetospheric object radius is much less than the ion gyroradius but greater than the electron gyroradius, the realized force is markedly reduced and electrons create force in proportion much greater than their relative mass with respect to ions as detailed in the [General kinematic model](https://bliptext.com/articles/) section where researchers report results from a compute intensive method that simulates individual particle interactions with the magnetic field source.\n# [magnetic sail | upload.wikimedia.org/wikipedia/commons/1/12/Magnetic///rope.svg]\n# [magnetic sail | upload.wikimedia.org/wikipedia/commons/0/05/VFPt///ringcurrent.svg]\n# [magnetic sail | upload.wikimedia.org/wikipedia/commons/6/6a/MFM///relative///force///vs///AU.png]\n\n# Modes of operation\nMagnetic sail modes of operation cover the mission profile and plasma environment (pe), such as the [solar wind](https://bliptext.com/articles/solar-wind), (sw) a planetary ionosphere (pi) or magnetosphere (pm), or the [interstellar medium](https://bliptext.com/articles/interstellar-medium) (ism).\nSymbolically equations in this article use the pe acronym as a subscript to generic variables, for example as described in this section the plasma mass density \\rho_{pe} and from the spacecraft point of view the apparent wind velocity u_{pe}.\n\n## Plasma mass density and velocity terminology and units\nA [plasma](https://bliptext.com/articles/plasma-physics) consists exclusively of charged particles that can interact with a magnetic or electric field.\nIt does not include neutral particles, such as neutrons.\natoms or molecules.The plasma mass density ρ used in magnetohydrodynamic models only require a weighted average mass density of charged particles that includes neutrons in the ion, while kinematic models use the values for each specific ion type and in some cases the parameters for electrons as well as detailed in the [Magnetohydrodynamic model](https://bliptext.com/articles/) section.\nThe velocity distribution of ions and electrons is another important parameter but often analyses use only the average velocity for the aggregate of particles in a plasma wind for a particular plasma environment (pe) is v_{pe}.\nThe apparent wind velocity u_{sc} as seen by a spacecraft traveling at velocity v_{sc} (positive meaning acceleration in the same direction as the wind and negative meaning deceleration opposite the wind direction) for a particular plasma environment (pe) is.\n\n## Acceleration/deceleration in a stellar plasma wind\nMany designs, analyses, simulations and experiments focus on using a magnetic sail in the [solar wind](https://bliptext.com/articles/solar-wind) plasma to accelerate a spacecraft away from the Sun.\nNear the Earth's orbit at 1 AU the plasma flows at velocity v_{sw} dynamically ranges from 250 to 750 km/s (typically 500), with a density ranging from 3 to 10 particles per cubic centimeter (typically 6) as reported by the NOAA real-time solar wind tracking web site Assuming that 8% of the solar wind is helium and the remainder hydrogen, the average solar wind plasma mass density at 1 AU is kg/m3 (typically 10−20 kg/m3).\nThe average plasma mass density of ions \\rho_{sw} decreases according to an [Inverse-square law](https://bliptext.com/articles/inverse-square-law) with the distance from the Sun as stated by Andrews/Zubrin and Borgazzi.\nThe velocity for values near the Sun is nearly constant, falling off slowly after 1 AU and then rapidly decreases at [heliopause](https://bliptext.com/articles/heliosphere).\n\n## Deceleration in interstellar medium (ISM)\nA spacecraft accelerated to very high velocities by other means, such as a fusion rocket or laser pushed lightsail, can decelerate even from relativistic velocities without onboard propellant by using a magnetic sail to create thrust (drag) against the interstellar medium plasma environment.\nAs shown in the se[c](https://bliptext.com/articles/speed-of-light)tion on [Magsail kinematic model (MKM)](https://bliptext.[c](https://bliptext.com/articles/speed-of-light)om/arti[c](https://bliptext.com/articles/speed-of-light)les/), feasible uses of this involve maximum velo[c](https://bliptext.com/articles/speed-of-light)ities below 10% [c](https://bliptext.com/articles/speed-of-light), taking de[c](https://bliptext.com/articles/speed-of-light)ades to de[c](https://bliptext.com/articles/speed-of-light)elerate, for total travel times on the order of a [c](https://bliptext.com/articles/speed-of-light)entury as des[c](https://bliptext.com/articles/speed-of-light)ribed in the magsail spe[c](https://bliptext.com/articles/speed-of-light)ifi[c](https://bliptext.com/articles/speed-of-light) designs se[c](https://bliptext.com/articles/speed-of-light)tion.\nOnly the magsail references consider deceleration in the ISM on approach to Alpha (\\alpha) Centauri, which as shown in the figure is separated by the [local bubble](https://bliptext.com/articles/local-bubble) and the [G-clouds](https://bliptext.com/articles/g-cloud) and the Solar System, which is moving at velocity v_{sun} and the local cloud is moving at velocity v_{L|C}.\nEstimates of the number of protons range between 0.005 and 0.5 cm−3 resulting in a plasma mass density kg/m3, which covers the range used by references in the magsail specific designs section.\nAs summarized in the magsail specific design section, Gros cited references indicating that regions of the G-clouds may be colder and have a low ion density.\nA typical value assumed for approach to Alpha Centauri is a proton [number density](https://bliptext.com/articles/number-density) n_{i} of 0.1 protons per cm3 corresponding to kg/m3.\nThe spacecraft velocity v_{sc} is much greater than the ISM velocity at the beginning of a deceleration maneuver so the apparent plasma wind velocity from the spacecraft's viewpoint s approximately.\nRadio emissions of [cyclotron radiation](https://bliptext.com/articles/cyclotron-radiation) due to interaction of charged particles in the interstellar medium as they spiral around the magnetic field lines of a magnetic sail would have a frequency of approximately kHz.\nThe Earth's ionosphere would prevent detection on the surface, but a space-based antenna could detect such emissions up to several thousands of light years away.\nDetection of such radiation could indicate activity of advanced extraterrestrial civilizations.\n# [magnetic sail | upload.wikimedia.org/wikipedia/commons/7/7c/The///Local///Interstellar///Cloud///and///neighboring///G-cloud///complex.svg]\n\n## In a planetary ionosphere\nA spacecraft approaching a planet with a significant upper atmosphere such as Saturn or Neptune could use a magnetic sail to decelerate by ionizing neutral atoms such that it behaves as a [low beta plasma](https://bliptext.com/articles/plasma-beta).\nThe plasma mass in a planetary ionosphere (pi) \\rho_{pi} is composed of multiple ion types and varies by altitude.\nThe spacecraft velocity v_{sc} is much greater than the planetary ionosphere velocity in a deceleration maneuver so the apparent plasma wind velocity is approximately at the beginning of a deceleration maneuver.\n\n## In a planetary magnetosphere\nInside or near a planetary [magnetosphere](https://bliptext.com/articles/magnetosphere), a magnetic sail can thrust against or be attracted to a planet's [magnetic field](https://bliptext.com/articles/magnetic-field) created by a [dynamo](https://bliptext.com/articles/dynamo-theory), especially in an [orbit](https://bliptext.com/articles/orbit) that passes over the planet's magnetic poles.\nWhen the magnetic sail and planet's magnetic field are in opposite directions an attractive force occurs and when the fields are in the same direction a repulsive force occurs, which is not stable and means to prevent the sail from flipping over is necessary.\nThe thrust that a magnetic sail delivers within a magnetosphere decreases with the fourth power of its distance from the planet's internal magnetic field.\nWhen close to a planet with a strong [magnetosphere](https://bliptext.com/articles/magnetosphere) such as [Earth](https://bliptext.com/articles/earth) or a [gas giant](https://bliptext.com/articles/gas-giant), the magnetic sail could generate more thrust by interacting with the [magnetosphere](https://bliptext.com/articles/magnetosphere) instead of the solar wind.\nWhen operating near a planetary or stellar magnetosphere the effect of that magnetic field must be considered if it is on the same order as the gravitational field.\nBy varying the magnetic sail's field strength and orientation a \"[perigee](https://bliptext.com/articles/perigee) kick\" can be achieved raising the altitude of the orbit's [apogee](https://bliptext.com/articles/apogee) higher and higher, until the magnetic sail is able to leave the planetary magnetosphere and catch the solar wind.\nThe same process in reverse can be used to lower or circularize the apogee of a magsail's orbit when it arrives at a destination planet with a magnetic field.\nIn theory, it is possible for a magnetic sail to launch directly from the surface of a planet near one of its magnetic poles, repelling itself from the planet's magnetic field.\nHowever, this requires the magnetic sail to be maintained in an \"unstable\" orientation.\nFurthermore, the magnetic sail must have extraordinarily strong magnetic fields for a launch from Earth, requiring superconductors supporting 80 times the current density of the best known high-temperature superconductors as of 1991.\nIn 2022 a spaceflight trial dubbed Jupiter Observing Velocity Experiment (JOVE) proposed using a [plasma magnet](https://bliptext.com/articles/plasma-magnet) to decelerate against the magnetosphere of Jupiter.\n\n# Physical principles\nPhysical principles involved include: interaction of magnetic fields with moving charged particles; an artificial magnetosphere model analogous to the [Earth's magnetosphere](https://bliptext.com/articles/earth-s-magnetosphere), MHD and kinematic mathematical models for interaction of an artificial magnetosphere with a plasma flow characterized by mass and number density and velocity, and performance measures; such as, force achieved, energy requirements and the mass of the magnetic sail system.\n\n## Magnetic field interaction with charged particles\nAn ion or electron with charge q in a plasma moving at velocity\nv\nin a [magnetic field](https://bliptext.com/articles/magnetic-field)\nB\nand [electric field](https://bliptext.com/articles/electric-field)\nE\nis treated as an idealized point charge in the [Lorentz force](https://bliptext.com/articles/lorentz-force).\nThis means that the force on an ion or electron is proportional to the product of their charge q and velocity component v perpendicular to the [magnetic field flux density](https://bliptext.com/articles/magnetic-field)\nB\n, in SI units as [teslas (T)](https://bliptext.com/articles/tesla-unit).\nA magnetic sail design introduces a magnetic field into a plasma flow which under certain conditions deflects the electrons and ions from their original trajectory with the particle's momentum transferred to the sail and hence the spacecraft thereby creating thrust.\nAn [electric sail](https://bliptext.com/articles/electric-sail) uses an electric field\nE\nthat under certain conditions interact with charged particles to create thrust.\n\n## Artificial magnetospheric model\nThe characteristics of the [Earth's magnetosphere](https://bliptext.com/articles/earth-s-magnetosphere) have been widely studied as a basis for magnetic sails.\nThe figure shows streamlines of charged particles from a plasma wind from the Sun (or a star) or an effective wind when decelerating in the ISM flowing from left to right.\nA source attached to a spacecraft generates a magnetic field.\nUnder certain conditions at the boundary where magnetic pressure equals the plasma wind kinetic pressure an artificial [bow shock](https://bliptext.com/articles/bow-shock) and [magnetopause](https://bliptext.com/articles/magnetopause) forms at a characteristic length L from the field source.\nThe ionized plasma wind particles create a [current sheet](https://bliptext.com/articles/current-sheet) along the magnetopause, which compresses the magnetic field lines facing the oncoming plasma wind by a factor of 2 at magnetopause as shown in Figure 2a.\nThe magnetopause deflects charged particles, which affects their streamlines and increases the density at magnetopause.\nA magnetospheric bubble or cavity forms that has very low density downstream from the magnetopause.\nUpstream from the magnetopause a [bow shock](https://bliptext.com/articles/bow-shock) develops.\nSimulation results often show the particle density through use of color with an example shown in the legend in the lower left.\nThis figure uses aspects of the general structure from Zubrin, Toivanen and Funaki and aspects of the plasma density from Khazanov and Cruz.\n# [Artificial magnetospheric model | upload.wikimedia.org/wikipedia/commons/b/bb/Artifical///Magnetospheric///model.jpg]\n\n## Magnetohydrodynamic model\nMagnetic sail designs operating in a plasma wind share a theoretical foundation based upon a [magnetohydrodynamic](https://bliptext.com/articles/magnetohydrodynamics) (MHD) model, sometimes called a fluid model, from [plasma physics](https://bliptext.com/articles/plasma-physics) for an artificially generated [magnetosphere](https://bliptext.com/articles/magnetosphere).\nUnder certain conditions, the plasma wind and the magnetic sail are separated by a [magnetopause](https://bliptext.com/articles/magnetopause) that blocks the charged particles, which creates a drag force that transfers (at least some) momentum to the magnetic sail, which then applies thrust to the attached spacecraft as described in Andrews/Zubrin, Cattell, Funaki, and Toivanen.\nA plasma environment has [fundamental parameters](https://bliptext.com/articles/plasma-parameters), and if a cited reference uses cgs units these should be converted to [SI units](https://bliptext.com/articles/si-units) as defined in the NRL plasma formulary, which this article uses as a reference for plasma parameter units not defined in [SI units](https://bliptext.com/articles/si-units).\nThe major parameters for plasma mass density are: the number of ions of type i per unit volume n_i the mass of each ion type accounting for isotopes m_i and the number of electrons n_e per unit volume each with [electron mass](https://bliptext.com/articles/electron-mass) m_e.\nAn average plasma mass density per unit volume for charged particles in a plasma environment pe (sw for stellar wind, pi for planetary ionosphere, im for interstellar medium) is expressed in equation form from [magnetohydrodynamics](https://bliptext.com/articles/magnetohydrodynamics) as.\nNote that this definition includes the mass of neutrons in an ion's nucleus.\nIn SI Units per unit volume is [cubic metre (m-3)](https://bliptext.com/articles/cubic-metre), mass is [kilogram](https://bliptext.com/articles/kilogram) (kg), and mass density is [kilogram](https://bliptext.com/articles/kilogram) per cubic metre (kg/m3).\nThe figure depicts the MHD model as described in Funaki and Djojodihardjo.\nStarting from the left a plasma wind in a plasma environment (e.g., stellar, ISM or an ionosphere) of effective velocity v_{pe} with density \\rho_{pe} encounters a spacecraft with time-varying velocity v_{sc} that is positive if accelerating and negative if decelerating.\nThe apparent plasma wind velocity from the spacecraft's viewpoint is.\nThe spacecraft and field source generate a [magnetic field](https://bliptext.com/articles/magnetic-field) that creates a magnetospheric bubble extending out to a magnetopause preceded by a bow shock that deflects electrons and ions from the plasma wind.\nAt the magnetopause the field source magnetic pressure equals the kinetic pressure of the plasma wind at a standoff shown at the bottom of the figure.\nThe characteristic length L is that of a circular sail of effective blocking area where is the effective magnetopause radius.\nUnder certain conditions the plasma wind pushing on the artificial magnetosphere bow shock and magnetopause creates a force F_w on the magnetic field source that is physically attached to the spacecraft so that at least part of the force F_w causes a force F_{sc} on the spacecraft, accelerating it when sailing downwind or decelerating when sailing into a headwind.\nUnder certain conditions and in some designs, some of the plasma wind force may be lost as indicated by F_{loss} on the right side.\nAll magnetic sail designs assume a standoff between plasma wind pressure p_w and magnetic pressure p_B with SI units of [Pascal (Pa, or N/m2)](https://bliptext.com/articles/pascal-unit) differing only in a constant coefficient C_{SO} as follows:\nwhere u is the apparent wind velocity and \\rho is the plasma mass density for a specific [plasma environment](https://bliptext.com/articles/magnetic-sail), B_{mp} the magnetic field flux density at [magnetopause](https://bliptext.com/articles/magnetopause), μ0 is the [vacuum permeability (N A-2)](https://bliptext.com/articles/vacuum-permeability) and C_{SO} is a constant that differs by reference as follows for C_{SO}=2 corresponding to p_w modeled as [dynamic pressure](https://bliptext.com/articles/dynamic-pressure) with no magnetic field compression, C_{SO}=1 for p_w modeled as [ram pressure](https://bliptext.com/articles/ram-pressure) with no magnetic field compression and C_{SO}=1/2 for p_w modeled as [ram pressure](https://bliptext.com/articles/ram-pressure) with magnetic field compression by a factor of 2 Equation can be solved to yield the required magnetic field B_{mp} that satisfies the pressure balance at [magnetopause](https://bliptext.com/articles/magnetopause) standoff as:\nThe force with SI Units of [Newtons (N)](https://bliptext.com/articles/newton-unit) derived by a magnetic sail for a plasma environment is determined from MHD equations as reported by principal researchers Funaki, Slough, Andrews and Zubrin, and Toivanen as follows:\nwhere C_d is a [coefficient of drag](https://bliptext.com/articles/coefficient-of-drag) determined by numerical analysis and/or simulation, \\rho u^2/2 is the wind pressure, and is the effective blocking area of the magnetic sail with magnetopause radius.\nNote that this equation has the same form as the [drag equation](https://bliptext.com/articles/drag-equation) in [fluid dynamics](https://bliptext.com/articles/fluid-dynamics).\nC_d is a function of [coil attack angle on thrust and steering angle](https://bliptext.com/articles/).\nThe [power (W)](https://bliptext.com/articles/power-physics) of the plasma wind is the product of velocity and a constant force\nwhere equation was used to derive the right side.\n# [Artificial Magnetosphere Model of Basic Magnetic Sail | upload.wikimedia.org/wikipedia/commons/e/e1/Artificial///Magnetosphere///Math///Model.jpg]\n\n### MHD applicability test\nAs summarized in the [overview](https://bliptext.com/articles/) section, an important condition for a magnetic sail to generate maximum force is that the magnetopause radius be on the order of an ion's radius of gyration.\nThrough analysis, numerical calculation, simulation and experimentation an important condition for a magnetic sail to generate significant force is the MHD applicability test, which states that the standoff distance L must be significantly greater than the ion [gyroradius](https://bliptext.com/articles/gyroradius), also called the Larmor radius or cyclotron radius:\nwhere m_i is the ion mass, v_\\perp is the velocity of a particle perpendicular to the magnetic field, |q| is the [elementary charge](https://bliptext.com/articles/elementary-charge) of the ion, B_x is the magnetic field flux density at the point of reference x and C_{Li} is a constant that differs by source with C_{Li}=1 and C_{Li}=2 . For example, in the solar wind with 5 ions/cm3 at 1 AU with m_i the [proton mass (kg)](https://bliptext.com/articles/proton-mass), = 400 km/s, B_x=B_{mp} = 36 nT with C_{SO}=0.5 from equation at magnetopause and C_{Li}=2 then r_g \\approx 72 km.\nThe MHD applicability test is the ratio r_g/L.\nThe figure plots C_d on the left vertical axis and lost thrust on the right vertical axis versus the ratio r_g/L.\nWhen r_g/L<1, C_d=3.6 is maximum, at, C_d=2.7, a decrease of 25% from the maximum and at , , a 45% decrease.\nAs r_g/L increases beyond one, C_d decreases meaning less thrust from the plasma wind transfers to the spacecraft and is instead lost to the plasma wind.\nIn 2004, Fujita published numerical analysis using a hybrid PIC simulation using a magnetic dipole model that treated electrons as a fluid and a kinematic model for ions to estimate the coefficient of drag C_d for a magnetic sail operating in the radial orientation resulting in the following approximate formula:The lost thrust is.\n# [magnetic sail | upload.wikimedia.org/wikipedia/commons/b/bf/MHD///applicability///test.jpg]\n\n### Coil attack angle effect on thrust and steering angle\nIn 2005 Nishida and others published results from numerical analysis of an MHD model for interaction of the solar wind with a magnetic field of current flowing in a coil that momentum is indeed transferred to the magnetic field produced by field source and hence to the spacecraft.\nThrust force derives from the momentum change of the solar wind, pressure by the solar wind on the magnetopause from equation and Lorentz force from currents induced in the magnetosphere interacting with the field source.\nThe results quantified the coefficient of drag, steering (i.e., thrust direction) angle with the solar wind, and torque generated as a function of attack angle (i.e., orientation) The figure illustrates how the attack (or coil tilt) angle \\alpha _t orientation of the coil creates a steering angle for the thrust vector and also torque imparted to the coil.\nAlso shown is the vector for the [interplanetary magnetic field](https://bliptext.com/articles/interplanetary-magnetic-field) (IMF), which at 1 AU varies with waves and other disturbances in the solar wind, known as [space weather](https://bliptext.com/articles/space-weather), and can significantly increase or decrease the thrust of a magnetic sail.\nFor a coil with radial orientation (like a Frisbee) the attack angle \\alpha _t= 0° and with axial orientation (like a parachute) \\alpha _t=90°.\nThe Nishida 2005 results reported a coefficient of drag C_d that increased non-linearly with attack angle from a minimum of 3.6 at \\alpha _t=0 to a maximum of 5 at \\alpha _t=90°.\nThe steering angle of the thrust vector is substantially less than the attack angle deviation from 45° due to the interaction of the magnetic field with the solar wind.\nTorque increases from \\alpha _t= 0° from zero at to a maximum at \\alpha _t=45° and then decreases to zero at \\alpha _t=90°.\nA number of magnetic sail design and other papers cite these results.\nIn 2012 Kajimura reported simulation results that covered two cases where MHD applicability occurs with r_g/L=1.125 and where a kinematic model is applicable r_g/L=0.125 to compute a coefficient of drag C_d and steering angle.\nAs shown in Figure 4 of that paper when MHD applicability occurs the results are similar in form to Nishida 2005 where the largest C_d occurs with the coil in an axial orientation.\nHowever, when the kinematic model applies, the largest C_d occurs with the coil in a radial orientation.\nThe steering angle is positive when MHD is applicable and negative when a kinematic model applies.\nThe 2012 Nishida and Funaki published simulation results for a coefficient of drag C_D, coefficient of lift C_L and a coefficient of moment C_M for a coil radius of R_c=100 km and magnetopause radius R_{mp}=500 km at 1 AU.\n# [Coil magnetic field orientation and forces | upload.wikimedia.org/wikipedia/commons/c/c1/Coil///magnetic///field///orientation///and///forces.jpg]\n\n## Magnetic field model\nIn a design, either the magnetic field source strength or the magnetopause radius the characteristic length must be chosen.\nA good approximation from Cattell and Toivanen for a magnetic field falloff rate for a distance from the field source to magnetopause starts with the equation:\nwhere B(R_0) is the magnetic field at a radius R_0 near the field source that falls off near the source as 1/r^3 as follows:\nwhere C_0 is a constant multiplying the [magnetic moment (A m2)](https://bliptext.com/articles/magnetic-moment) \\mathbf m to make B(r) match a target value at r>>R_0.\nWhen far from the field source, a magnetic dipole is a good approximation and choosing the above value of B(R_0) with C_0 =2 near the field source was used by Andrews and Zubrin.\nThe [Amperian loop model](https://bliptext.com/articles/magnetic-moment) for the magnetic moment is, where I_c is the current in [amperes (A)](https://bliptext.com/articles/ampere) and is the surface area for a coil (loop) of radius R_c.\nAssuming that and substituting the expression for the magnetic moment \\mathbf m into equation yields the following:\nWhen the magnetic field flux density B(R_0) is specified, substituting B_{mp} from the pressure balance analysis from equation into the above and solving for R_{mp} yields the following:\nThis is the expression for L when f_o=3 with C_{SO}=1/2 and C_{SO}=2 and is the same form as the [magnetopause distance of the Earth](https://bliptext.com/articles/magnetopause).\nEquation shows directly how a decreased falloff rate f_o dramatically increases the effective sail area for a given field source magnetic moment \\mathbf m and B_{mp} determined from the pressure balance equation.\nSubstituting this into equation yields the plasma wind force as a function of falloff rate f_o, plasma density \\rho, coil radius, coil current I_c and plasma wind velocity u as follows:\nusing equation for B(R_0) and equation for B_{mp}.\nThis is the same expression as equation (10b) when f_o=3 and C_{SO}=1/2 and and the right hand side from equation (20) specifically applied to M2P2 with other numerical coefficients grouped into the C_d term.\nNote that force increases as falloff rate decreases.\nFor the solar wind case, substituting into and using the function for the [solar wind plasma mass density](https://bliptext.com/articles/), with a_\\odot the distance from the sun in Astronomical units (AU) results in the following expression: where , the effective sail blocking area.\nThis equation explicitly shows the relationship upon solar wind plasma mass density as a function of distance from the Sun a_\\odot.\nFor the case f_o=1 the expansion of the magnetopause radius R_{mp} exactly matches the decreasing value of exactly as the distance from the Sun a_\\odot increases, resulting in constant force and hence constant acceleration inside the heliosphere.\nNote that C_{f_o} includes the term, which means that as f_o increases that the magnetic field near the field source B(R_0) must increase to maintain the same force as compared with a smaller value of f_o.\nThe example in the [overview](https://bliptext.com/articles/) section set C_{f_o}=1, u=1, R_0=1, and S=1 so that the force at a_\\odot=1 was equal to 1 for all values of f_o at 1 AU.\n\n## General kinematic model\nWhen the [MHD applicability test](https://bliptext.com/articles/) of r_g/L <1 then a kinematic simulation model more accurately predicts force transferred from the plasma wind to the spacecraft.\nIn this case the effective sail blocking area A_e < S=\\pi L^2.\nThe left axis of the figure is for plots of magnetic sail force versus characteristic length L. The solid black line plots the MHD model force F_{MHD} from equation.\nThe green line shows the value of ion gyroradius r_g \\approx 72 km at 1 AU from equation.\nThe dashed blue line plots the hybrid MHD/kinematic model from equation from Fujita04.\nThe red dashed line plots a curve fit to simulation results from Ashida14.\nAlthough a good fit for these parameters, the curve fit range of this model does not cover some relevant examples.\nAdditional simulation results from Hajiwara15 are shown for the MHD and kinematic model as single data points as indicated in the legend.\nThese models are all in close agreement.\nThe kinematic models predict less force than predicted by the MHD model.\nIn other words, the fraction T_{loss} of thrust force predicted by the MHD model is lost when r_g/L<1 as plotted on the right axis.\nThe solid blue and red lines show T_{loss} for Fujita04 and Ashida18 respectively, indicating that operation with L less than 10% of r_g will have significant loss.\nOther factors in a specific magnetic sail design may offset this loss for values of L1 then the Injection of a high-velocity, high-density plasma into a magnetosphere as proposed in [M2P2](https://bliptext.com/articles/) freezes the motion of a magnetic field into the plasma flow and was believed to inflate the magnetosphere.\nHowever 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, similar to another criticism of M2P2.\nAn alternative is to reduce the plasma injection velocity and density to result in \\beta_k<1 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.\nIn 2013 Funaki and others published simulation and theoretical results regarding how characteristics of the injected plasma affected thrust gain through creation of an equatorial ring current.\nThey defined thrust gain for MPS as : the ratio of the force generated by low beta plasma injection F_{MPS} divided by that of a pure magnetic sail F_{mag} from equation with f_o =3 and C_{SO}=0.5 for r_g\\leq L or from equation for r_g>L.\nThey reported G_{MPS} 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.\nSimulations revealed that optimum thrust gain occurred for \\beta_k<1 and.\nIn 2014 Arita, Nishida and Funaki published simulation results 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 G_{MPS} of 3.77 with.\nThey 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.\nIn 2014 Ashida and others documented Particle In Cell (PIC) simulation results for a kinematic model for cases where r_g >> L where MHD is not applicable.\nEquation (12) of their study included the additional force of the injected plasma jet F_{jet} consisting of momentum and static pressure of ions and electrons and defined thrust gain as, which differs from the definition of a term by the same name in other studies.\nIt represents the gain of MPS over that of simply adding the magnetic sail force and the plasma injection jet force.\nFor the values cited in the conclusion, is 7.5 in the radial orientation.\nSince a number of results were published by different authors at different times, the figure summarizes the reported thrust gain G_{MPS} versus magnetosphere size (or characteristic length L) with the source indicated in the legend as follows for simulation results Arita14, Ashida14, Funaki13, and Kajimura10.\nSimulation 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.\nA 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.\nThe MHD applicability test of equation for the solar wind is L \\approx72 km. Therefore, the estimated force of the MPS is that of equation multiplied by the empirically determined thrust gain G_{MPS} from the figure multiplied by the percentage thrust loss T_{loss} from equation\nFor example, using solar wind parameters \\rho=8x10−21 kg/m3 and u=500 km/s then r_g=72 km and B_{mp}=4x10−8 T. With L=105 m for f_o=3 then r_g < L and 11% from equation.\nThe magnetic field only force with a coil radius of R_c=6,300 m and coil current I_c=1.6x106 A yields B_0=1.6x10−4 T from equation and with C_d=5 the magnetic force only is 175 N from equation.\nDetermining 4 from the figure at L=105 m as the multiplier for the magnetic-only force then the MPS force 700 N.\nSince MPS injects ionized gas at a rate of m_{in} that can be viewed as a propellant it has a [specific impulse](https://bliptext.com/articles/specific-impulse) where g_0 is the acceleration of [Earth's gravity](https://bliptext.com/articles/earth-s-gravity).\nFunaki and Arita stated m_{in}=0.31 kg/day.\nTherefore I_{sp}=28,325 s per newton of thrust force.\nThe equivalent exhaust velocity is 278 km/s per newton of thrust force.\nIn 2015 Kajimura and others published simulation results for thrust performance with plasma injected by a magnetic nozzle, a technology used in [VASIMR](https://bliptext.com/articles/variable-specific-impulse-magnetoplasma-rocket).\nThey reported a thrust gain G_{MPS} of 24 when the ion [gyroradius](https://bliptext.com/articles/gyroradius) r_g (see equation ) was comparable to the characteristic length L, at the boundary of the [MHD applicability test](https://bliptext.com/articles/).\nThe optimal result occurred with a thermal with some decrease for higher values of thermal beta.\nIn 2015 Hagiwara and Kajimura published experimental thrust performance test results with plasma injection using a [magnetoplasmadynamic thruster](https://bliptext.com/articles/magnetoplasmadynamic-thruster) (aka MPD thruster or MPD Arcjet) in a single direction opposite the solar wind direction and a coil with the axial orientation.\nThis meant that F_{jet} provided additional propulsive force.\nDensity plots explicitly show the increased plasma density upwind of the bow shock originating from the MPD thruster.\nThey reported that showing how MPS inflated the magnetic field to create more thrust than the magnetic sail alone plus that of the < >.\nThe conclusion of the experiment was that the thrust gain G_{MPS} was approximately 12 for a scaled characteristic length of L = 60 km. In the above figure, note the significant improvement in thrust gain at L = 60 km.as compared with only plasma injection.\nIn this example, using solar wind parameters \\rho=8x10−21 kg/m3 and u=500 km/s then r_g=72 km and B_{mp}=4x10−8 T. With L=60 km for f_o=3 then and 28% from equation.\nThe magnetic field only force with a coil radius of R_c=2,900 m and coil current I_c=1.6x106 A yields B_0=3.5x10−4 T from equation and with C_d=5 the magnetic force only is 51 N from equation.\nGiven G_{MPS}=12 as the multiplier for the magnetic only force then the MPS force 611 N.\nIn 2017 Ueno published a design proposing use of multiple coils to generate a more complex magnetic field to increase thrust production.\nIn 2020 Murayama and others published additional experimental results for a multi-pole MPD thruster.\nIn 2017 Djojodihardjo published a conceptual design using MPS for a small (~500 kg) Earth observation satellite.\nIn 2020 Peng and others published MHD simulation results for a magnetic dipole with plasma injection operating in [Low Earth orbit](https://bliptext.com/articles/low-earth-orbit) at 500 km within the Earth's [Ionosphere](https://bliptext.com/articles/ionosphere) where the ion number density is approximately 1011 m−3.\nAs reported in Figure 3, the magnetic field strength initially falls off as 1/r and then approaches 1/r2 at larger distances from the dipole.\nThe radius of the artificial mini-magnetosphere could extend up to 200 m for this scenario.\nThey 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.\n# [Magnetoplasma sail (MPS) schematic | upload.wikimedia.org/wikipedia/commons/0/0f/Magnetoplasma///sail///schematic.jpg]\n# [Summary of MPS thrust gain results | upload.wikimedia.org/wikipedia/commons/0/02/Summary///of///MPS///thrust///gain///results.jpg]\n\n## Plasma magnet (PM)\nThe plasma magnet (PM) sail design introduced a different approach to generate a static magnetic dipole as illustrated in the figure.\nAs shown in the detailed view on the right the field source is two relatively small crossed perpendicularly oriented antenna coils each of radius R_c (m), each carrying a sinusoidal [alternating current](https://bliptext.com/articles/alternating-current) (AC) with the total current of I_c (A) generated by an onboard power supply.\nThe AC current applied to each coil is out of phase by 90° and consequently generates a [rotating magnetic field](https://bliptext.com/articles/rotating-magnetic-field) (RMF) with [rotational frequency (s-1)](https://bliptext.com/articles/rotational-speed) chosen that is fast enough that positive ions do not rotate but the less massive electrons rotate at this speed.\nThe figure illustrates rotation using color coded contours of constant magnetic strength, not magnetic field lines.\nIn order to inflate the magnetospheric bubble the thermal [plasma beta](https://bliptext.com/articles/plasma-beta) \\beta_t must be high and initially a plasma injection may be necessary, analogous to inflating a balloon when small and internal tension is high.\nAfter 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 Rmp, which is orders of magnitude larger than the coil radius Rc (figure not drawn to scale).\nSee RMDCartoon.avi for an animation of this effect.\nThe induced current disc carries a direct current I_{cd} orders of magnitude larger than the input alternating current I_c and forms a static dipole magnetic field oriented perpendicular to the current disc reaching a standoff balance with the plasma wind pressure at distance at the magnetopause boundary according to the [MHD model of an artificial magnetosphere](https://bliptext.com/articles/).\nThe magnetic field falloff rate was assumed in 2001 and 2006 to be f_o =1.\nHowever, as described by Khazanov in 2003 and restated by Slough, Kirtley and Pancotti in 2011 and Kirtley and Slough in a 2012 NIAC report that f_o=2 as demanded by [conservation of magnetic flux](https://bliptext.com/articles/alfv-n-s-theorem).\nSeveral [MPS](https://bliptext.com/articles/) studies concluded that f_o is closer to 2.\nThe falloff rate f_o is a critical parameter in the determination of performance.\nThe RMF-induced rotating disc of electrons has [current density (A m-2)](https://bliptext.com/articles/current-density) j_\\theta(r) at distance r from the antenna for f_o=1 and for f_o=2, which states that flux conservation requires this falloff rate, consistent with a criticism of M2P2 by Cattell as follows:\nwhere B(R_0) T is the magnetic field flux density at radius m near the antenna coils.\nNote that the current density is highest at r=R_0 and falls off at a rate of f_o+1.\nA critical condition for the plasma magnet design provides a lower bound on the RMF frequency 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:\nwhere \\omega_{ci} is the [ion gyrofrequency (s-1)](https://bliptext.com/articles/ion-cyclotron-resonance) in the RMF near the antenna coils, Z is charge number of the ion, e is the [elementary charge](https://bliptext.com/articles/elementary-charge), and m_i kg is the (average) mass of the ion(s).\nSpecifying the magnetic field near the coils at radius R_0 is critical since this is where the current density is greatest.\nChoosing a magnetic field at magnetopause yields a lower value of but ions closer to the coils will rotate.\nAnother condition is that be small enough such that collisions are extremely unlikely.\nThe required power to generate the RMF P_{RMF} is derived by integrating the product of the square of the current density from equation and the resistivity of the plasma \\eta_p from R_0 to R_{mp} with the result as follows:\nwhere \\eta_p is the [Spitzer resistivity (W m)](https://bliptext.com/articles/spitzer-resistivity) of the plasma of ~1.2x10−3 T_e^{-3/2} where T_e is the electron temperature assumed to be 15 eV, the same result for f_o=1 and for f_o=2.\nStarting with the definition of plasma wind force from equation, noting that P_w=F_w u rearranging and recognizing that equation gives the solution for B(R_0), which can be substituted and then using equation for B_{mp} yields the following expression\nwhich when multiplied by C_d/2 with C_SO=1 is the same as for f_o=1 Note that solution for P_{RMF} and R_0 must also satisfy equation, to which the comments following regarding a \"tremendous leverage of power\" do not address.\nNote that a number of the examples cited in assume a magnetopause radius R_{mp} that do not meet the MHD applicability test of equation.\nFrom the [definition of power in physics](https://bliptext.com/articles/power-physics) a constant force is power divided by velocity, the force generated by the plasma magnet (PM) sail is as follows from equation Comparing the above with Equation not the dependence on plasma mass density \\rho is of the same form.\nNote from Equation that as the falloff rate f_o increases that the force derived from the plasma wind decreases, or to maintain the same force P_{RMF} and/or R_0 must increase to maintain the same force F_{PM}.\nEquation gives the mass for each physical coil of radius R_c m. Since the RMF requires alternating current and semiconductors are not efficient at higher frequencies, aluminum was specified with mass density \\delta_c = 2,700 kg/m3.\nEstimates of the coil mass are optimistic by a factor of 4 \\pi since only one coil was sized and the coil circumference was specified as R_c instead of 2 \\pi R_c.\nThe coil [resistance](https://bliptext.com/articles/electrical-resistivity-and-conductivity) is the product of coil material [resistivity (Ω m)](https://bliptext.com/articles/electrical-resistivity-and-conductivity) \\eta_c (e.g., ~3x10−8 Ωm for aluminum) and the coil length 2 \\pi R_c divided by the coil wire cross sectional area where r_c is the radius of the coil wire as follows:\nSome additional power must compensate for resistive loss but it is orders of magnitude less than P_{RMF}.\nThe peak current carried by a coil is specified by the RMF power and coil resistance from the definition of [electrical power in physics](https://bliptext.com/articles/power-physics) as follows:\nThe current induced in the disc by the RMF I_{ic} is the integral of the current density j(r,f_o) from equation on the surface of the disc with inner radius R_c and outer radius R_{MP} with result:\nthe same result for f_o=1.\nLaboratory experiments 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.\nSince the scale of a terrestrial experiment is limited, simulations or a flight trial was recommended.\nSome of these concepts adapted to an ionospheric plasma environment were carried on in the [plasma magnetoshell](https://bliptext.com/articles/) design.\nIn 2022 Freeze, Greason and others published a detailed design for a [plasma magnet](https://bliptext.com/articles/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).\nThis design employed a pair of superconducting coils each with radius R_c of 9 m, an alternating current of I_c of 112 A with and a falloff rate of f_o=1.\nA transit time to Jupiter of 25 days was reported for a 21 kg spacecraft design launched in a 16 U Cubesat format.\nUsing f_o=1 creates very optimistic performance numbers, but since Slough changed this to f_o=2 in 2011 and 2012, the case of f_o=1 is not compared in this article.\nAn example for f_o=2 using solar wind parameters \\rho=8x10−21 kg/m3, u=500 km/s then r_g=72 km and B_{mp}=4x10−8 T with R_{mp}=105 m results in r_g < L where MHD applicability occurs.\nWith a coil radius of R_c=1,000 m yields B_0=4x10−4 T from equation.\nThe required RMF power from equation is 13 kW with a required AC coil current I_c=10 A from equation resulting in an induced current of I_{ic}=2 kA from equation.\nWith C_d=5 the plasma magnet force from equation is 197 N. The magnetic force only for the above parameters is 2.8 N from equation and therefore the plasma magnet thrust gain is 71.\nThe [performance comparison](https://bliptext.com/articles/) section gives and optimistic estimate using constant acceleration for f_o=2 results in a transit time of ~100 days.\n# [Plasma magnet principles of operation | upload.wikimedia.org/wikipedia/commons/c/c0/Plasma///magnet///principles///of///operation.jpg]\n\n## Plasma magnetoshell (PMS)\nA 2011 paper by Slough and others and a 2012 NIAC report by Kirtley investigated use of the plasma magnet technology with 1/r2 magnetic field falloff rate for use in the ionosphere of a planet as a braking mechanism in an approach dubbed Plasma magnetoshell (PMS).\nThe magnetoshell creates drag by ionizing neutral atoms in a planet's ionosphere then magnetically deflecting them.\nA tether attaching the plasma magnet coils to the spacecraft transfers momentum such that orbital insertion occurs.\nAnalytical models, laboratory demonstrations and mission profiles to Neptune and Mars were described.\nIn 2017, Kelly described using a single-coil magnet with 1/r3 magnetic field falloff rate and more experimental results.\nIn 2019 Kelly and Little published simulation results for magnetoshell performance scaling.\nA magnet with radius R_c=1 m was tethered to a spacecraft with batteries for 1,000 seconds of operation (longer than aerocapture designs).\nThe simulations assumed a magnet mass M_c=1,000 kg and total magnetoshell system mass of 1,623 kg, suitable for a [Cassini–Huygens](https://bliptext.com/articles/cassini-huygens) or [Juno](https://bliptext.com/articles/juno-spacecraft) size orbiter.\nThe planet's mass and atmosphere atomic composition and density determine a threshold velocity where magnetoshell operation is feasible.\nSaturn and Neptune have a hydrogen atmosphere and a threshold velocity of approximately 22 km/s.\nIn a Neptune mission a \\Delta v=6 km is required for a 5,000 kg spacecraft and must average 50 kN for the maneuver duration.\nThe model overestimates performance for N2 (Earth, Titan) and CO2 (Venus, Mars) atmospheres since multiple ion species are created and more complex interactions occur.\nFurthermore, the relatively lower mass of Venus and Mars reduces the threshold velocity below that of feasible magnetoshell operation.\nThe paper states that [aerocapture](https://bliptext.com/articles/aerocapture) technologies are mature enough for these mission profiles.\nIn 2021, Kelly and Little published further details for use of drag-modulated plasma aerocapture (DMPA) that when compared to Adaptable Deployable Entry and Placement Technology (ADEPT) for drag-modulated aerocapture (DMA) to Neptune that could deliver 70% higher orbiter mass and experience 30% lower stagnation heating.\n\n## Beam powered magsail (BPM)\nA [beam-powered](https://bliptext.com/articles/beam-powered-propulsion) of M2P2 variant, [MagBeam](https://bliptext.com/articles/magnetized-beamed-plasma-propulsion) was proposed in 2011.\nIn this design a magnetic sail is used with [beam-powered propulsion](https://bliptext.com/articles/beam-powered-propulsion), by using a high-power [particle accelerator](https://bliptext.com/articles/particle-accelerator) to fire a beam of charged particles at the spacecraft.\nThe magsail would deflect this beam, transferring momentum to the vehicle, that could provide higher acceleration than a solar sail driven by a [laser](https://bliptext.com/articles/laser), but a charged particle beam would disperse in a shorter distance than a [laser](https://bliptext.com/articles/laser) due to the electrostatic repulsion of its component particles.\nThis 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](https://bliptext.com/articles/jordin-kare).\n\n# Performance comparison\nThe table below compares [performance measures](https://bliptext.com/articles/) for the magnetic sail designs with the following parameters for the solar wind (sw) at 1 AU: velocity u_{sw}= 500 km/s, number density n_i= 5x106 m−3, ion mass m_i = 1.67x10−27 kg a [proton mass](https://bliptext.com/articles/proton-mass), resulting in mass density = 8.4x10−21 kg/m3, and coefficient of drag C_d=5 where applicable.\nEquation gives the magnetic field at magnetopause as B_{mp}≈ 36 nT, equation gives the ion gyroradius r_g≈ 72 km for C_{Li}=2.\nTable entries in boldface are from a cited source as described in the following:\nEquation determines force for the [Magsail (MS)](https://bliptext.com/articles/) divided by the Freeland correction factor 3.1, equation defines the force for the [plasma magnet (PM)](https://bliptext.com/articles/) with the assumed magnetic field falloff rate f_o=2.\nThe force for the magnetic sail alone F_{mag} is from equation.\nThrust gain G_T for the [magneto plasma sail (MPS)](https://bliptext.com/articles/) is the simulation and/or experimentally determined value with force defined equation to account for thrust loss due to operation in a kinematic region.\nThe last column headed MPS+MPD adds a [magnetoplasma dynamic thruster](https://bliptext.com/articles/magnetoplasmadynamic-thruster) (MPD) that has a higher thrust gain as determined by experiment and simulation.\nFurther details are in the section for the specific design.\nFor designs other than MPS and MPS+MPD, the thrust gain G_T is the achieved force from the first row divided by the force of a magnetic sail alone in the second row.\nThe magnetopause distance and the coil radius R_c are design parameters.\nEquation with r=R_{mp} defines the magnetic field near the coil(s) as.\nThe superconducting coil designs used a critical current density J_c=2x106 A/m to account for warmer temperatures in the solar system.\nThe plasma magnet uses AC power for the rotating magnetic field, PRMF as specified in Equation and cannot use a superconducting coil and assumed an aluminum coil with material density \\delta_c = 2,700 kg/m3 and coil wire radius r_c=5 mm.\nAll other designs assumed a superconducting coil with material density \\delta_c =6,500 kg/m3, coil wire radius r_c=5 mm, and critical current 1.6 x106 A, above which the coil becomes a normal conductor.\nThe magnetopause distance and coil radius R_c for superconducting-coil based designs were adjusted to meet this critical current constraint.\nThe values for the plasma magnet used a value of R_c for f_o=2 selected to minimize time to velocity and distance.\nThe MPS values for and R_c were chosen to match the thrust gain from simulation and scaled experimental results and meet the superconducting-coil critical current constraint.\nEquation gives the physical coil mass M_c(phy) assuming a coil wire radius r_c=5 mm. Equation gives the plasma magnet alternating current I_c.\nEquation gives the direct current I_c with C_0=2 for all other designs.\nThe plasma magnet RMF uses the input alternating current I_c (kA) to rotate electrons in captured plasma to create an induced direct current disc carrying I_{ic} kA as defined in equation.\nSuperconducting coils do not require continuous power (except possibly for cooling); however, the plasma magnet design does, as specified in equation.\nAn estimate for the plasma magnet power supply mass assumes ~3 kg/W for [nuclear power in space](https://bliptext.com/articles/nuclear-power-in-space).\nOther mass was assumed to be 10 tonnes for MS and 1 tonne for PM and MPS. Acceleration a is the thrust force F from the first row divided by the total mass (coil plus other).\nAn optimistic approximation is constant acceleration a, for which the time to reach a target velocity V of 10% of the solar wind velocity is and time to cover a specified distance D ≈ 7.8x108 km (approximate distance from Earth to Jupiter) is.\nFor comparison purposes the time for a [Hohmann transfer](https://bliptext.com/articles/delta-v-budget) from Earth orbit to Jupiter orbit is 2.7 years (almost 1,000 days) but that would allow orbital insertion whereas a magnetic sail would do a flyby unless the magnetosphere and gravity of Jupiter could provide deceleration.\nAnother comparison is the [New Horizons](https://bliptext.com/articles/new-horizons) interplanetary space probe with a 30 kg payload that flew by Jupiter after 405 days on its way to Pluto.\nThe best time to velocity T_V and distance T_D performance occurs for the PM and MPS designs due primarily to much reduced coil and other mass.\nAs described in the [M2P2 section](https://bliptext.com/articles/), several criticisms asserted that the falloff rate f_o=1 was questionable and hence it was not included in this table.\nSimulations and experiments as described in the [MPS section](https://bliptext.com/articles/) showed that f_o=2 is valid with injection of plasma to inflate the magnetic field in a manner similar to M2P2.\nAs described in the [PM section](https://bliptext.com/articles/), plasma is not injected but instead captured to achieve a falloff rate of f_o=2, with calculations assuming f_o=1 being very optimistic.\nThe classic [Magsail (MS)](https://bliptext.com/articles/) design generates the most thrust force and has considerable mass but still has relatively good time performance.\nParameters for the other designs were chosen to yield comparable time performance subject to the constraints previously described.\nAs described above and further detailed in the section for the respective design, this article contains the equations and parameters to compute performance estimates for different parameter choices.\n\n# Criticisms\nIn 1994 Vulpetti published a critical review regarding viability of space propulsion based on solar wind momentum flux.\nThe paper highlighted technology challenges in terms of the magnetic field source, energy required and interaction between the solar wind and the spacecraft's magnetic field, summarizing that these issues were not insurmountable.\nThe major unresolved issue is spacecraft and mission design that account for the potentially highly variable solar wind velocity and plasma density that could complicate maneuvers by a spacecraft employing magnetic sail technology.\nSome means of modulating thrust is necessary.\nIf the mission objective is to rapidly escape the solar system then the paper states that this is less of an issue.\nIn 2006, Bolonkin published a paper that questioned the theoretical viability of a Magsail and described common mistakes.\nEquation (2) states that the magnetic field of electrons rotating in the large coil was greater than and opposed the magnetic field generated by the current in the coil and hence no thrust would result.\nIn 2014 Vulpetti published a rebuttal that summarized plasma properties, in particular the fact that plasma is quasi-neutral and noted in equation (B1) that the Bolonkin paper equation (2) assumed that the plasma had a large net negative electrical charge.\nThe plasma charge varies statistically over short intervals and the maximum value has negligible impact on Magsail performance.\nFurthermore, he argued that observations by many spacecraft have observed compression of a magnetic field by dynamic (or ram) pressure that did not depend on particle charges.\nIn 2017, Gros published results that differed from prior magsail work.\nA major result was the [Magsail kinetic model](https://bliptext.com/articles/) of equation that is a curve fit to numerical analysis of proton trajectories impacted by a large current carrying superconducting coil.\nThe curve fit scaling relation for the effective sail area A(v) was logarithmic cubed with argument with I the loop current, I_G the curve fit parameter, v the ship velocity and c the speed of light.\nThis differed from the power law scaling of prior work.\nThe Gros paper could not trace back this difference to underlying physical arguments and noted that the results are inconsistent, stating that the source for these discrepancies was unclear.\nAppendix B questioned whether a bow shock will form if the initial spacecraft velocity v_0 is large, for example for deceleration after interstellar travel, since the predicted effective sail area A_G(v) is small in this case.\nOne difference is that this analysis used the coil radius R_c for computation of the ion gyroradius as compared with prior work use of the magnetopause radius R_{mp}\n\n# Fictional uses in popular culture\nMagnetic sails have become a popular [trope](https://bliptext.com/articles/trope-literature) in many works of [science fiction](https://bliptext.com/articles/science-fiction) although the [solar sail](https://bliptext.com/articles/solar-sail) is more popular:\nAlthough not referred to as a \"magnetic sail\", the concept was used in the novel [Encounter with Tiber](https://bliptext.com/articles/encounter-with-tiber) by [Buzz Aldrin](https://bliptext.com/articles/buzz-aldrin) and [John Barnes](https://bliptext.com/articles/john-barnes-author) as a braking mechanism to decelerate starships from relativistic speed.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:54:42.743Z","updated_at":"2025-01-25T17:54:42.743Z","search_vector":"'magnet':1 'sail':2","search_content":" \n \n\n# \nA **magnetic sail** is a proposed method of [spacecraft propulsion](https://bliptext.com/articles/spacecraft-propulsion) where an onboard magnetic field source interacts with a [plasma](https://bliptext.com/articles/plasma-physics) wind (e.g., the [solar wind](https://bliptext.com/articles/solar-wind)) to form an [artificial [magnetosphere](https://bliptext.com/articles/magnetosphere)](https://bliptext.com/articles/) (similar to Earth's [magnetosphere](https://bliptext.com/articles/magnetosphere)) that acts as a sail, transferring force from the wind to the spacecraft requiring little to no propellant as detailed for each proposed **magnetic sail** design in this article.\nThe animation and the following text summarize the [magnetic sail physical principles](https://bliptext.com/articles/magnetic-sail) involved.\nThe spacecraft's [magnetic field](https://bliptext.com/articles/magnetic-field) source, represented by the purple dot, generates a [magnetic field](https://bliptext.com/articles/magnetic-field), shown as expanding black circles.\nUnder conditions summarized in the [overview](https://bliptext.com/articles/) section, this field creates a magnetosphere whose leading edge is a [magnetopause](https://bliptext.com/articles/magnetopause) and a [bow shock](https://bliptext.com/articles/bow-shock) composed of [charged particles](https://bliptext.com/articles/charged-particle) captured from the wind by the magnetic field, as shown in blue, which deflects subsequent [charged particles](https://bliptext.com/articles/charged-particle) from the plasma wind coming from the left.\nSpecific attributes of the [artificial magnetosphere](https://bliptext.com/articles/) around the spacecraft for a specific design significantly affect performance as summarized in the overview section.\nA [magnetohydrodynamic model](https://bliptext.com/articles/magnetic-sail) (verified by computer simulations and laboratory experiments) predicts that the interaction of the artificial magnetosphere with the oncoming plasma wind creates an effective sail blocking area that transfers force as shown by a sequence of labeled arrows from the plasma wind, to the spacecraft's magnetic field, to the spacecraft's field source, which accelerates the spacecraft in the same direction as the plasma wind.\nThese concepts apply to all [proposed magnetic sail system designs](https://bliptext.com/articles/), with the difference how the design generates the magnetic field and how efficiently the field source creates the [artificial magnetosphere](https://bliptext.com/articles/) described above.\nThe [History of concept](https://bliptext.com/articles/) section summarizes key aspects of the proposed designs and relationships between them as background.\nThe cited references are technical with many equations and in order to make the information more accessible, this article first describes in text (and illustrations where available) beginning in the overview section and prior to each design, section or groups of equations and plots intended for the technically oriented reader.\nThe beginning of each proposed design section also contains a summary of the important aspects so that a reader can skip the equations for that design.\nThe differences in the designs determine [performance measures](https://bliptext.com/articles/), such as the mass of the field source and necessary power, which in turn determine force, mass and hence acceleration and velocity that enable a [performance comparison](https://bliptext.com/articles/) between magnetic sail designs at the end of this article.\nA [comparison with other spacecraft propulsion methods](https://bliptext.com/articles/spacecraft-propulsion) includes some magnetic sail designs where the reader can click on the column headers to compare magnetic sail performance with other propulsion methods.\nThe following observations result from this comparison: magnetic sail designs have insufficient thrust to launch from Earth, thrust (drag) for deceleration for the [magsail](https://bliptext.com/articles/) in the interstellar medium is relatively large, and both the [magsail](https://bliptext.com/articles/) and [magnetoplasma sail](https://bliptext.com/articles/) both have significant thrust for travel away from Earth using the force from the solar wind.\n\n# History of concept\nAn overview of many of the magnetic sail proposed designs with illustrations from the references was published in 2018 by Djojodihardjo.\nThe earliest method proposed by Andrews and Zubrin in 1988, dubbed the [magsail](https://bliptext.com/articles/), has the significant advantage of requiring no propellant and is thus a form of [field propulsion](https://bliptext.com/articles/field-propulsion) that can operate indefinitely.\nA drawback of the magsail design was that it required a large (50–100 km radius) superconducting loop carrying large currents with a mass on the order of 100 tonnes.\nThe magsail design also described [modes of operation](https://bliptext.com/articles/) for interplanetary transfers, thrusting against a planetary [ionosphere](https://bliptext.com/articles/ionosphere) or [magnetosphere](https://bliptext.com/articles/magnetosphere), escape from low Earth orbit as well as deceleration of an interstellar craft over decades after being initially accelerated by other means, for example.\na [fusion rocket](https://bliptext.com/articles/fusion-rocket), to a significant fraction of light speed, with a more detailed design published in 2000.\nIn 2015, Freeland validated most of the initial magsail analysis, but determined that thrust predictions were optimistic by a factor of 3.1 due to a numerical integration error.\nSubsequent designs proposed and analyzed means to significantly reduce mass.\nThese designs require little to modest amounts of exhausted propellant and can thrust for years.\nAll proposed designs describe thrust from solar wind outwards from the Sun.\nIn 2000, Winglee and Slough proposed a [Mini-Magnetospheric Plasma Propulsion (M2P2)](https://bliptext.com/articles/) design that injected low energy plasma into a much smaller coil with much lower mass that required low power.\nSimulations predicted impressive performance relative to mass and required power; however, a number of critiques raised issues: that the assumed magnetic field falloff rate was optimistic and that thrust was dramatically overestimated.\nStarting in 2003, Funaki and others published a series of theoretical, simulation and experimental investigations at JAXA in collaboration with Japanese universities addressing some of the issues from criticisms of M2P2 and named their approach the [MagnetoPlasma Sail (MPS)](https://bliptext.com/articles/).\nIn 2011, Funaki and Yamakawa authored a chapter in a book that is a good reference for magnetic sail theory and concepts.\nMPS research resulted in many published papers that advanced the understanding of [physical principles for magnetic sails](https://bliptext.com/articles/).\nBest performance occurred when the injected plasma had a lower density and velocity than considered in M2P2.\nThrust gain was computed as compared with performance with a magnetic field only in 2013 and 2014.\nInvestigations and experiments continued reporting increased thrust experimentally and numerically considering use of a [Magnetoplasmadynamic thruster](https://bliptext.com/articles/magnetoplasmadynamic-thruster) (aka MPD Arc jet in Japan)in 2015, multiple antenna coils in 2019, and a multi-pole MPD thruster in 2020.\nSlough published in 2004 and 2006 a method to generate the static magnetic dipole for a magnetic sail in a design called the [Plasma magnet (PM)](https://bliptext.com/articles/) that was described as an [AC induction motor](https://bliptext.com/articles/induction-motor) turned inside out.\nA pair of small perpendicularly oriented coils acted as the stator powered by an alternating current to generate a [rotating magnetic field (RMF)](https://bliptext.com/articles/rotating-magnetic-field) that analysis predicted and laboratory experiments demonstrated that a current disc formed as the rotor outside the stator.\nThe current disk formed from electrons captured from the plasma wind, therefore requiring little to no plasma injection.\nPredictions of substantial improvements in terms of reduced coil size (and hence mass) and markedly lower power requirements for significant thrust hypothesized the same optimistic magnetic field falloff rate as assumed for M2P2.\nIn 2022, a spaceflight trial dubbed Jupiter Observing Velocity Experiment (JOVE) proposed using a [plasma magnet](https://bliptext.com/articles/plasma-magnet) based sail for a spacecraft named Wind Rider using the solar wind to accelerate away from a point near Earth and decelerate against the magnetosphere of Jupiter.\nA 2012, study by Kirtley and Slough investigated using the plasma magnet technology to use plasma in a planetary ionosphere as a braking mechanism and was called the Plasma Magnetoshell.\nThis paper restated the magnetic field falloff rate to the value suggested in the critiques of M2P2 that dramatically reduces analytical predicted performance.\nInitial missions targeted deceleration in the ionosphere of Mars.\nKelly and Little in 2019 published simulation results using a multi-turn coil and not the plasma magnet showed that the magnetoshell was viable for orbital insertion asy Mars, Jupiter, Neptune and Uranus and in 2021 showed that it was more efficient than [aerocapture](https://bliptext.com/articles/aerocapture) for Neptune.\nIn 2021, Zhenyu Yang and others published an analysis, numerical calculations and experimental verification for a propulsion system that was a combination of the magnetic sail and the [electric sail](https://bliptext.com/articles/electric-sail) called an electromagnetic sail.\nA superconducting magsail coil augmented by an [electron gun](https://bliptext.com/articles/electron-gun) at the coil's center generates an electric field as in an electric sail that deflects positive ions in the plasma wind thereby providing additional thrust, which could reduce overall system mass.\n\n# Overview\nThe [Modes of operation](https://bliptext.com/articles/) section describes the important parameters of [plasma particle density and wind velocity](https://bliptext.com/articles/) in conjunction with a use case for:\nThe [Physical principles](https://bliptext.com/articles/) section details aspects of how charged particles in a plasma wind interact with a magnetic field and conditions that determine how much thrust force results on the spacecraft in terms of particle's behavior in a plasma wind, as well as the form and magnitude of the magnetic field related to conditions within the magnetosphere that differ for the proposed designs.\n[Charged particles](https://bliptext.com/articles/charged-particle) such as electrons, protons and ions travel in straight lines in a vacuum in the absence of a magnetic field.\nAs shown in the illustration in the presence of a [magnetic field](https://bliptext.com/articles/magnetic-field) shown in green, charged particles gyrate in circular arcs with blue indicating positively charged particles (e.g., protons) and red indicating electrons.\nThe particle's [gyroradius](https://bliptext.com/articles/gyroradius) is proportional to the ratio of the particle's [momentum](https://bliptext.com/articles/momentum) (product of mass and velocity) over the magnetic field.\nAt 1 [Astronomical Unit (AU)](https://bliptext.com/articles/astronomical-unit), the distance from the Sun to the Earth, the gyroradius of a [proton](https://bliptext.com/articles/proton) is ~72 km and since a [proton](https://bliptext.com/articles/proton) is ~1,836 times the mass of an [electron](https://bliptext.com/articles/electron), the gyroradius of an [electron](https://bliptext.com/articles/electron) is ~40 m with the illustration not drawn to scale.\nFor the magsail [deceleration in the interstellar medium (ISM)](https://bliptext.com/articles/) mode of operation the velocity is a significant fraction of light speed, for example 5% c, the gyroradius is ~ 500 km for protons and ~280 m for electrons.\nWhen the magsail magnetopause radius is much less than the proton gyroradius the [magsail kinematic model](https://bliptext.com/articles/) by Gros in 2017, which considered only protons, predicts a marked reduction in thrust force for initial ship velocity greater than 10% c prior to deceleration.\nWhen the magnetosphere radius is much greater than the spacecraft's magnetic field source radius, all proposed designs, except for the magsail, use a [magnetic dipole](https://bliptext.com/articles/magnetic-dipole) approximation for an [Amperian loop](https://bliptext.com/articles/magnetic-moment) shown in the center of the illustration with the X indicating current flowing into the page and the dot indicating current flowing out of the page.\nThe illustration shows the resulting [magnetic field lines](https://bliptext.com/articles/magnetic-field) and their direction, where the closer spacing of lines indicates a stronger field.\nSince the magsail uses a large superconducting coil that has a radius on the same order as the magnetosphere the details of that design use the [magsail MHD model](https://bliptext.com/articles/) employing the [Biot–Savart law](https://bliptext.com/articles/biot-savart-law) that predicts stronger magnetic fields near and inside the coil than the dipole model.\nA [Lorentz force](https://bliptext.com/articles/lorentz-force) occurs only for the portion of a charged particle's velocity at a right angle to the magnetic field lines and this constitutes the magnetic force depicted in the summary animation.\nElectrically neutral particles, such as neutrons, atoms and molecules are unaffected by a magnetic field.\nA condition for [applicability of magnetohydrodynamic (MHD)](https://bliptext.com/articles/) theory, which models charged particles as fluid flows, is that to achieve maximum force the radius of the [artificial magnetosphere](https://bliptext.com/articles/) be on the same order as the ion gyroradius for the plasma environment for a [particular mode of operation](https://bliptext.com/articles/).\nAnother important condition is how the proposed design affects the magnetic field falloff rate inside the magnetosphere, which impacts the field source mass and power requirements.\nFor a radial distance r from the spacecraft's magnetic field source in a vacuum the magnetic field falls off as 1/r^{f_o}, where f_o is the falloff rate.\nClassic [magnetic dipole](https://bliptext.com/articles/magnetic-dipole) theory covers the case of f_o=3 as used in the magsail design.\nWhen plasma is injected and/or captured near the field source, the magnetic field falls off at a rate of, a topic that has been a subject of much research, criticism and differs between designs and has changed over time for the plasma magnet.\nThe M2P2 and plasma magnet designs initially assumed f_o=1 that as shown in numerical examples summarized at the end of the corresponding design sections predicted a very large performance gain.\nSeveral researchers independently created a [magnetic field model](https://bliptext.com/articles/) where and asserted that an f_o=2 falloff rate was the best achievable.\nIn 2011 the [plasma magnet](https://bliptext.com/articles/) author changed the falloff rate f_o from 1 to 2 and that is the value used for the [plasma magnet](https://bliptext.com/articles/) for performance comparison in this article.\nThe [magnetoplasma sail (MPS)](https://bliptext.com/articles/) design is an evolution of the M2P2 concept that has been extensively documented, numerically analyzed and simulated and reported a falloff rate f_o between 1.5 and 2.\nThe falloff rate f_o has a significant impact on performance or the mode of operation [accelerating away from the Sun](https://bliptext.com/articles/) where the mass density of ions in the plasma decreases according to an [Inverse-square law](https://bliptext.com/articles/inverse-square-law) with distance from the Sun (e.g., AU) increases.\nThe illustration shows in a [semi-log plot](https://bliptext.com/articles/semi-log-plot) the impact of falloff rate f_o on relative force F from Equation versus distance from the Sun ranging from 1 to 20 AU, the approximate distance of Neptune.\nThe distance to Jupiter is approximately 5 AU.\nConstant force independent of distance from the Sun for f_o=1 is stated in several plasma magnet references, for example Slough and Freeze and results from the effective increase in sail blocking area to exactly offset reduced plasma mass density as a magnetic sail spacecraft accelerates in response to the plasma wind force away from the Sun.\nAs seen from the illustration the impact of falloff rate f_o on force, and therefore acceleration, becomes grerater as distance from the Sun increases.\nAt scales where the artificial magnetospheric object radius is much less than the ion gyroradius but greater than the electron gyroradius, the realized force is markedly reduced and electrons create force in proportion much greater than their relative mass with respect to ions as detailed in the [General kinematic model](https://bliptext.com/articles/) section where researchers report results from a compute intensive method that simulates individual particle interactions with the magnetic field source.\n# [magnetic sail | upload.wikimedia.org/wikipedia/commons/1/12/Magnetic///rope.svg]\n# [magnetic sail | upload.wikimedia.org/wikipedia/commons/0/05/VFPt///ringcurrent.svg]\n# [magnetic sail | upload.wikimedia.org/wikipedia/commons/6/6a/MFM///relative///force///vs///AU.png]\n\n# Modes of operation\nMagnetic sail modes of operation cover the mission profile and plasma environment (pe), such as the [solar wind](https://bliptext.com/articles/solar-wind), (sw) a planetary ionosphere (pi) or magnetosphere (pm), or the [interstellar medium](https://bliptext.com/articles/interstellar-medium) (ism).\nSymbolically equations in this article use the pe acronym as a subscript to generic variables, for example as described in this section the plasma mass density \\rho_{pe} and from the spacecraft point of view the apparent wind velocity u_{pe}.\n\n## Plasma mass density and velocity terminology and units\nA [plasma](https://bliptext.com/articles/plasma-physics) consists exclusively of charged particles that can interact with a magnetic or electric field.\nIt does not include neutral particles, such as neutrons.\natoms or molecules.The plasma mass density ρ used in magnetohydrodynamic models only require a weighted average mass density of charged particles that includes neutrons in the ion, while kinematic models use the values for each specific ion type and in some cases the parameters for electrons as well as detailed in the [Magnetohydrodynamic model](https://bliptext.com/articles/) section.\nThe velocity distribution of ions and electrons is another important parameter but often analyses use only the average velocity for the aggregate of particles in a plasma wind for a particular plasma environment (pe) is v_{pe}.\nThe apparent wind velocity u_{sc} as seen by a spacecraft traveling at velocity v_{sc} (positive meaning acceleration in the same direction as the wind and negative meaning deceleration opposite the wind direction) for a particular plasma environment (pe) is.\n\n## Acceleration/deceleration in a stellar plasma wind\nMany designs, analyses, simulations and experiments focus on using a magnetic sail in the [solar wind](https://bliptext.com/articles/solar-wind) plasma to accelerate a spacecraft away from the Sun.\nNear the Earth's orbit at 1 AU the plasma flows at velocity v_{sw} dynamically ranges from 250 to 750 km/s (typically 500), with a density ranging from 3 to 10 particles per cubic centimeter (typically 6) as reported by the NOAA real-time solar wind tracking web site Assuming that 8% of the solar wind is helium and the remainder hydrogen, the average solar wind plasma mass density at 1 AU is kg/m3 (typically 10−20 kg/m3).\nThe average plasma mass density of ions \\rho_{sw} decreases according to an [Inverse-square law](https://bliptext.com/articles/inverse-square-law) with the distance from the Sun as stated by Andrews/Zubrin and Borgazzi.\nThe velocity for values near the Sun is nearly constant, falling off slowly after 1 AU and then rapidly decreases at [heliopause](https://bliptext.com/articles/heliosphere).\n\n## Deceleration in interstellar medium (ISM)\nA spacecraft accelerated to very high velocities by other means, such as a fusion rocket or laser pushed lightsail, can decelerate even from relativistic velocities without onboard propellant by using a magnetic sail to create thrust (drag) against the interstellar medium plasma environment.\nAs shown in the se[c](https://bliptext.com/articles/speed-of-light)tion on [Magsail kinemati[c](https://bliptext.com/articles/speed-of-light) model (MKM)](https://bliptext.[c](https://bliptext.com/articles/speed-of-light)om/arti[c](https://bliptext.com/articles/speed-of-light)les/), feasible uses of this involve maximum velo[c](https://bliptext.com/articles/speed-of-light)ities below 10% [c](https://bliptext.com/articles/speed-of-light), taking de[c](https://bliptext.com/articles/speed-of-light)ades to de[c](https://bliptext.com/articles/speed-of-light)elerate, for total travel times on the order of a [c](https://bliptext.com/articles/speed-of-light)entury as des[c](https://bliptext.com/articles/speed-of-light)ribed in the magsail spe[c](https://bliptext.com/articles/speed-of-light)ifi[c](https://bliptext.com/articles/speed-of-light) designs se[c](https://bliptext.com/articles/speed-of-light)tion.\nOnly the magsail references consider deceleration in the ISM on approach to Alpha (\\alpha) Centauri, which as shown in the figure is separated by the [local bubble](https://bliptext.com/articles/local-bubble) and the [G-clouds](https://bliptext.com/articles/g-cloud) and the Solar System, which is moving at velocity v_{sun} and the local cloud is moving at velocity v_{L|C}.\nEstimates of the number of protons range between 0.005 and 0.5 cm−3 resulting in a plasma mass density kg/m3, which covers the range used by references in the magsail specific designs section.\nAs summarized in the magsail specific design section, Gros cited references indicating that regions of the G-clouds may be colder and have a low ion density.\nA typical value assumed for approach to Alpha Centauri is a proton [number density](https://bliptext.com/articles/number-density) n_{i} of 0.1 protons per cm3 corresponding to kg/m3.\nThe spacecraft velocity v_{sc} is much greater than the ISM velocity at the beginning of a deceleration maneuver so the apparent plasma wind velocity from the spacecraft's viewpoint s approximately.\nRadio emissions of [cyclotron radiation](https://bliptext.com/articles/cyclotron-radiation) due to interaction of charged particles in the interstellar medium as they spiral around the magnetic field lines of a magnetic sail would have a frequency of approximately kHz.\nThe Earth's ionosphere would prevent detection on the surface, but a space-based antenna could detect such emissions up to several thousands of light years away.\nDetection of such radiation could indicate activity of advanced extraterrestrial civilizations.\n# [magnetic sail | upload.wikimedia.org/wikipedia/commons/7/7c/The///Local///Interstellar///Cloud///and///neighboring///G-cloud///complex.svg]\n\n## In a planetary ionosphere\nA spacecraft approaching a planet with a significant upper atmosphere such as Saturn or Neptune could use a magnetic sail to decelerate by ionizing neutral atoms such that it behaves as a [low beta plasma](https://bliptext.com/articles/plasma-beta).\nThe plasma mass in a planetary ionosphere (pi) \\rho_{pi} is composed of multiple ion types and varies by altitude.\nThe spacecraft velocity v_{sc} is much greater than the planetary ionosphere velocity in a deceleration maneuver so the apparent plasma wind velocity is approximately at the beginning of a deceleration maneuver.\n\n## In a planetary magnetosphere\nInside or near a planetary [magnetosphere](https://bliptext.com/articles/magnetosphere), a magnetic sail can thrust against or be attracted to a planet's [magnetic field](https://bliptext.com/articles/magnetic-field) created by a [dynamo](https://bliptext.com/articles/dynamo-theory), especially in an [orbit](https://bliptext.com/articles/orbit) that passes over the planet's magnetic poles.\nWhen the magnetic sail and planet's magnetic field are in opposite directions an attractive force occurs and when the fields are in the same direction a repulsive force occurs, which is not stable and means to prevent the sail from flipping over is necessary.\nThe thrust that a magnetic sail delivers within a magnetosphere decreases with the fourth power of its distance from the planet's internal magnetic field.\nWhen close to a planet with a strong [magnetosphere](https://bliptext.com/articles/magnetosphere) such as [Earth](https://bliptext.com/articles/earth) or a [gas giant](https://bliptext.com/articles/gas-giant), the magnetic sail could generate more thrust by interacting with the [magnetosphere](https://bliptext.com/articles/magnetosphere) instead of the solar wind.\nWhen operating near a planetary or stellar magnetosphere the effect of that magnetic field must be considered if it is on the same order as the gravitational field.\nBy varying the magnetic sail's field strength and orientation a \"[perigee](https://bliptext.com/articles/perigee) kick\" can be achieved raising the altitude of the orbit's [apogee](https://bliptext.com/articles/apogee) higher and higher, until the magnetic sail is able to leave the planetary magnetosphere and catch the solar wind.\nThe same process in reverse can be used to lower or circularize the apogee of a magsail's orbit when it arrives at a destination planet with a magnetic field.\nIn theory, it is possible for a magnetic sail to launch directly from the surface of a planet near one of its magnetic poles, repelling itself from the planet's magnetic field.\nHowever, this requires the magnetic sail to be maintained in an \"unstable\" orientation.\nFurthermore, the magnetic sail must have extraordinarily strong magnetic fields for a launch from Earth, requiring superconductors supporting 80 times the current density of the best known high-temperature superconductors as of 1991.\nIn 2022 a spaceflight trial dubbed Jupiter Observing Velocity Experiment (JOVE) proposed using a [plasma magnet](https://bliptext.com/articles/plasma-magnet) to decelerate against the magnetosphere of Jupiter.\n\n# Physical principles\nPhysical principles involved include: interaction of magnetic fields with moving charged particles; an artificial magnetosphere model analogous to the [Earth's magnetosphere](https://bliptext.com/articles/earth-s-magnetosphere), MHD and kinematic mathematical models for interaction of an artificial magnetosphere with a plasma flow characterized by mass and number density and velocity, and performance measures; such as, force achieved, energy requirements and the mass of the magnetic sail system.\n\n## Magnetic field interaction with charged particles\nAn ion or electron with charge q in a plasma moving at velocity\nv\nin a [magnetic field](https://bliptext.com/articles/magnetic-field)\nB\nand [electric field](https://bliptext.com/articles/electric-field)\nE\nis treated as an idealized point charge in the [Lorentz force](https://bliptext.com/articles/lorentz-force).\nThis means that the force on an ion or electron is proportional to the product of their charge q and velocity component v perpendicular to the [magnetic field flux density](https://bliptext.com/articles/magnetic-field)\nB\n, in SI units as [teslas (T)](https://bliptext.com/articles/tesla-unit).\nA magnetic sail design introduces a magnetic field into a plasma flow which under certain conditions deflects the electrons and ions from their original trajectory with the particle's momentum transferred to the sail and hence the spacecraft thereby creating thrust.\nAn [electric sail](https://bliptext.com/articles/electric-sail) uses an electric field\nE\nthat under certain conditions interact with charged particles to create thrust.\n\n## Artificial magnetospheric model\nThe characteristics of the [Earth's magnetosphere](https://bliptext.com/articles/earth-s-magnetosphere) have been widely studied as a basis for magnetic sails.\nThe figure shows streamlines of charged particles from a plasma wind from the Sun (or a star) or an effective wind when decelerating in the ISM flowing from left to right.\nA source attached to a spacecraft generates a magnetic field.\nUnder certain conditions at the boundary where magnetic pressure equals the plasma wind kinetic pressure an artificial [bow shock](https://bliptext.com/articles/bow-shock) and [magnetopause](https://bliptext.com/articles/magnetopause) forms at a characteristic length L from the field source.\nThe ionized plasma wind particles create a [current sheet](https://bliptext.com/articles/current-sheet) along the magnetopause, which compresses the magnetic field lines facing the oncoming plasma wind by a factor of 2 at magnetopause as shown in Figure 2a.\nThe magnetopause deflects charged particles, which affects their streamlines and increases the density at magnetopause.\nA magnetospheric bubble or cavity forms that has very low density downstream from the magnetopause.\nUpstream from the magnetopause a [bow shock](https://bliptext.com/articles/bow-shock) develops.\nSimulation results often show the particle density through use of color with an example shown in the legend in the lower left.\nThis figure uses aspects of the general structure from Zubrin, Toivanen and Funaki and aspects of the plasma density from Khazanov and Cruz.\n# [Artificial magnetospheric model | upload.wikimedia.org/wikipedia/commons/b/bb/Artifical///Magnetospheric///model.jpg]\n\n## Magnetohydrodynamic model\nMagnetic sail designs operating in a plasma wind share a theoretical foundation based upon a [magnetohydrodynamic](https://bliptext.com/articles/magnetohydrodynamics) (MHD) model, sometimes called a fluid model, from [plasma physics](https://bliptext.com/articles/plasma-physics) for an artificially generated [magnetosphere](https://bliptext.com/articles/magnetosphere).\nUnder certain conditions, the plasma wind and the magnetic sail are separated by a [magnetopause](https://bliptext.com/articles/magnetopause) that blocks the charged particles, which creates a drag force that transfers (at least some) momentum to the magnetic sail, which then applies thrust to the attached spacecraft as described in Andrews/Zubrin, Cattell, Funaki, and Toivanen.\nA plasma environment has [fundamental parameters](https://bliptext.com/articles/plasma-parameters), and if a cited reference uses cgs units these should be converted to [SI units](https://bliptext.com/articles/si-units) as defined in the NRL plasma formulary, which this article uses as a reference for plasma parameter units not defined in [SI units](https://bliptext.com/articles/si-units).\nThe major parameters for plasma mass density are: the number of ions of type i per unit volume n_i the mass of each ion type accounting for isotopes m_i and the number of electrons n_e per unit volume each with [electron mass](https://bliptext.com/articles/electron-mass) m_e.\nAn average plasma mass density per unit volume for charged particles in a plasma environment pe (sw for stellar wind, pi for planetary ionosphere, im for interstellar medium) is expressed in equation form from [magnetohydrodynamics](https://bliptext.com/articles/magnetohydrodynamics) as.\nNote that this definition includes the mass of neutrons in an ion's nucleus.\nIn SI Units per unit volume is [cubic metre (m-3)](https://bliptext.com/articles/cubic-metre), mass is [kilogram](https://bliptext.com/articles/kilogram) (kg), and mass density is [kilogram](https://bliptext.com/articles/kilogram) per cubic metre (kg/m3).\nThe figure depicts the MHD model as described in Funaki and Djojodihardjo.\nStarting from the left a plasma wind in a plasma environment (e.g., stellar, ISM or an ionosphere) of effective velocity v_{pe} with density \\rho_{pe} encounters a spacecraft with time-varying velocity v_{sc} that is positive if accelerating and negative if decelerating.\nThe apparent plasma wind velocity from the spacecraft's viewpoint is.\nThe spacecraft and field source generate a [magnetic field](https://bliptext.com/articles/magnetic-field) that creates a magnetospheric bubble extending out to a magnetopause preceded by a bow shock that deflects electrons and ions from the plasma wind.\nAt the magnetopause the field source magnetic pressure equals the kinetic pressure of the plasma wind at a standoff shown at the bottom of the figure.\nThe characteristic length L is that of a circular sail of effective blocking area where is the effective magnetopause radius.\nUnder certain conditions the plasma wind pushing on the artificial magnetosphere bow shock and magnetopause creates a force F_w on the magnetic field source that is physically attached to the spacecraft so that at least part of the force F_w causes a force F_{sc} on the spacecraft, accelerating it when sailing downwind or decelerating when sailing into a headwind.\nUnder certain conditions and in some designs, some of the plasma wind force may be lost as indicated by F_{loss} on the right side.\nAll magnetic sail designs assume a standoff between plasma wind pressure p_w and magnetic pressure p_B with SI units of [Pascal (Pa, or N/m2)](https://bliptext.com/articles/pascal-unit) differing only in a constant coefficient C_{SO} as follows:\nwhere u is the apparent wind velocity and \\rho is the plasma mass density for a specific [plasma environment](https://bliptext.com/articles/magnetic-sail), B_{mp} the magnetic field flux density at [magnetopause](https://bliptext.com/articles/magnetopause), μ0 is the [vacuum permeability (N A-2)](https://bliptext.com/articles/vacuum-permeability) and C_{SO} is a constant that differs by reference as follows for C_{SO}=2 corresponding to p_w modeled as [dynamic pressure](https://bliptext.com/articles/dynamic-pressure) with no magnetic field compression, C_{SO}=1 for p_w modeled as [ram pressure](https://bliptext.com/articles/ram-pressure) with no magnetic field compression and C_{SO}=1/2 for p_w modeled as [ram pressure](https://bliptext.com/articles/ram-pressure) with magnetic field compression by a factor of 2 Equation can be solved to yield the required magnetic field B_{mp} that satisfies the pressure balance at [magnetopause](https://bliptext.com/articles/magnetopause) standoff as:\nThe force with SI Units of [Newtons (N)](https://bliptext.com/articles/newton-unit) derived by a magnetic sail for a plasma environment is determined from MHD equations as reported by principal researchers Funaki, Slough, Andrews and Zubrin, and Toivanen as follows:\nwhere C_d is a [coefficient of drag](https://bliptext.com/articles/coefficient-of-drag) determined by numerical analysis and/or simulation, \\rho u^2/2 is the wind pressure, and is the effective blocking area of the magnetic sail with magnetopause radius.\nNote that this equation has the same form as the [drag equation](https://bliptext.com/articles/drag-equation) in [fluid dynamics](https://bliptext.com/articles/fluid-dynamics).\nC_d is a function of [coil attack angle on thrust and steering angle](https://bliptext.com/articles/).\nThe [power (W)](https://bliptext.com/articles/power-physics) of the plasma wind is the product of velocity and a constant force\nwhere equation was used to derive the right side.\n# [Artificial Magnetosphere Model of Basic Magnetic Sail | upload.wikimedia.org/wikipedia/commons/e/e1/Artificial///Magnetosphere///Math///Model.jpg]\n\n### MHD applicability test\nAs summarized in the [overview](https://bliptext.com/articles/) section, an important condition for a magnetic sail to generate maximum force is that the magnetopause radius be on the order of an ion's radius of gyration.\nThrough analysis, numerical calculation, simulation and experimentation an important condition for a magnetic sail to generate significant force is the MHD applicability test, which states that the standoff distance L must be significantly greater than the ion [gyroradius](https://bliptext.com/articles/gyroradius), also called the Larmor radius or cyclotron radius:\nwhere m_i is the ion mass, v_\\perp is the velocity of a particle perpendicular to the magnetic field, |q| is the [elementary charge](https://bliptext.com/articles/elementary-charge) of the ion, B_x is the magnetic field flux density at the point of reference x and C_{Li} is a constant that differs by source with C_{Li}=1 and C_{Li}=2 . For example, in the solar wind with 5 ions/cm3 at 1 AU with m_i the [proton mass (kg)](https://bliptext.com/articles/proton-mass), = 400 km/s, B_x=B_{mp} = 36 nT with C_{SO}=0.5 from equation at magnetopause and C_{Li}=2 then r_g \\approx 72 km.\nThe MHD applicability test is the ratio r_g/L.\nThe figure plots C_d on the left vertical axis and lost thrust on the right vertical axis versus the ratio r_g/L.\nWhen r_g/L >R_0.\nWhen far from the field source, a magnetic dipole is a good approximation and choosing the above value of B(R_0) with C_0 =2 near the field source was used by Andrews and Zubrin.\nThe [Amperian loop model](https://bliptext.com/articles/magnetic-moment) for the magnetic moment is, where I_c is the current in [amperes (A)](https://bliptext.com/articles/ampere) and is the surface area for a coil (loop) of radius R_c.\nAssuming that and substituting the expression for the magnetic moment \\mathbf m into equation yields the following:\nWhen the magnetic field flux density B(R_0) is specified, substituting B_{mp} from the pressure balance analysis from equation into the above and solving for R_{mp} yields the following:\nThis is the expression for L when f_o=3 with C_{SO}=1/2 and C_{SO}=2 and is the same form as the [magnetopause distance of the Earth](https://bliptext.com/articles/magnetopause).\nEquation shows directly how a decreased falloff rate f_o dramatically increases the effective sail area for a given field source magnetic moment \\mathbf m and B_{mp} determined from the pressure balance equation.\nSubstituting this into equation yields the plasma wind force as a function of falloff rate f_o, plasma density \\rho, coil radius, coil current I_c and plasma wind velocity u as follows:\nusing equation for B(R_0) and equation for B_{mp}.\nThis is the same expression as equation (10b) when f_o=3 and C_{SO}=1/2 and and the right hand side from equation (20) specifically applied to M2P2 with other numerical coefficients grouped into the C_d term.\nNote that force increases as falloff rate decreases.\nFor the solar wind case, substituting into and using the function for the [solar wind plasma mass density](https://bliptext.com/articles/), with a_\\odot the distance from the sun in Astronomical units (AU) results in the following expression: where , the effective sail blocking area.\nThis equation explicitly shows the relationship upon solar wind plasma mass density as a function of distance from the Sun a_\\odot.\nFor the case f_o=1 the expansion of the magnetopause radius R_{mp} exactly matches the decreasing value of exactly as the distance from the Sun a_\\odot increases, resulting in constant force and hence constant acceleration inside the heliosphere.\nNote that C_{f_o} includes the term, which means that as f_o increases that the magnetic field near the field source B(R_0) must increase to maintain the same force as compared with a smaller value of f_o.\nThe example in the [overview](https://bliptext.com/articles/) section set C_{f_o}=1, u=1, R_0=1, and S=1 so that the force at a_\\odot=1 was equal to 1 for all values of f_o at 1 AU.\n\n## General kinematic model\nWhen the [MHD applicability test](https://bliptext.com/articles/) of r_g/L 1 then the Injection of a high-velocity, high-density plasma into a magnetosphere as proposed in [M2P2](https://bliptext.com/articles/) freezes the motion of a magnetic field into the plasma flow and was believed to inflate the magnetosphere.\nHowever 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, similar to another criticism of M2P2.\nAn alternative is to reduce the plasma injection velocity and density to result in \\beta_k L.\nThey reported G_{MPS} 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.\nSimulations revealed that optimum thrust gain occurred for \\beta_k > L where MHD is not applicable.\nEquation (12) of their study included the additional force of the injected plasma jet F_{jet} consisting of momentum and static pressure of ions and electrons and defined thrust gain as, which differs from the definition of a term by the same name in other studies.\nIt represents the gain of MPS over that of simply adding the magnetic sail force and the plasma injection jet force.\nFor the values cited in the conclusion, is 7.5 in the radial orientation.\nSince a number of results were published by different authors at different times, the figure summarizes the reported thrust gain G_{MPS} versus magnetosphere size (or characteristic length L) with the source indicated in the legend as follows for simulation results Arita14, Ashida14, Funaki13, and Kajimura10.\nSimulation 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.\nA 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.\nThe MHD applicability test of equation for the solar wind is L \\approx72 km. Therefore, the estimated force of the MPS is that of equation multiplied by the empirically determined thrust gain G_{MPS} from the figure multiplied by the percentage thrust loss T_{loss} from equation\nFor example, using solar wind parameters \\rho=8x10−21 kg/m3 and u=500 km/s then r_g=72 km and B_{mp}=4x10−8 T. With L=105 m for f_o=3 then r_g .\nThe conclusion of the experiment was that the thrust gain G_{MPS} was approximately 12 for a scaled characteristic length of L = 60 km. In the above figure, note the significant improvement in thrust gain at L = 60 km.as compared with only plasma injection.\nIn this example, using solar wind parameters \\rho=8x10−21 kg/m3 and u=500 km/s then r_g=72 km and B_{mp}=4x10−8 T. With L=60 km for f_o=3 then and 28% from equation.\nThe magnetic field only force with a coil radius of R_c=2,900 m and coil current I_c=1.6x106 A yields B_0=3.5x10−4 T from equation and with C_d=5 the magnetic force only is 51 N from equation.\nGiven G_{MPS}=12 as the multiplier for the magnetic only force then the MPS force 611 N.\nIn 2017 Ueno published a design proposing use of multiple coils to generate a more complex magnetic field to increase thrust production.\nIn 2020 Murayama and others published additional experimental results for a multi-pole MPD thruster.\nIn 2017 Djojodihardjo published a conceptual design using MPS for a small (~500 kg) Earth observation satellite.\nIn 2020 Peng and others published MHD simulation results for a magnetic dipole with plasma injection operating in [Low Earth orbit](https://bliptext.com/articles/low-earth-orbit) at 500 km within the Earth's [Ionosphere](https://bliptext.com/articles/ionosphere) where the ion number density is approximately 1011 m−3.\nAs reported in Figure 3, the magnetic field strength initially falls off as 1/r and then approaches 1/r2 at larger distances from the dipole.\nThe radius of the artificial mini-magnetosphere could extend up to 200 m for this scenario.\nThey 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.\n# [Magnetoplasma sail (MPS) schematic | upload.wikimedia.org/wikipedia/commons/0/0f/Magnetoplasma///sail///schematic.jpg]\n# [Summary of MPS thrust gain results | upload.wikimedia.org/wikipedia/commons/0/02/Summary///of///MPS///thrust///gain///results.jpg]\n\n## Plasma magnet (PM)\nThe plasma magnet (PM) sail design introduced a different approach to generate a static magnetic dipole as illustrated in the figure.\nAs shown in the detailed view on the right the field source is two relatively small crossed perpendicularly oriented antenna coils each of radius R_c (m), each carrying a sinusoidal [alternating current](https://bliptext.com/articles/alternating-current) (AC) with the total current of I_c (A) generated by an onboard power supply.\nThe AC current applied to each coil is out of phase by 90° and consequently generates a [rotating magnetic field](https://bliptext.com/articles/rotating-magnetic-field) (RMF) with [rotational frequency (s-1)](https://bliptext.com/articles/rotational-speed) chosen that is fast enough that positive ions do not rotate but the less massive electrons rotate at this speed.\nThe figure illustrates rotation using color coded contours of constant magnetic strength, not magnetic field lines.\nIn order to inflate the magnetospheric bubble the thermal [plasma beta](https://bliptext.com/articles/plasma-beta) \\beta_t must be high and initially a plasma injection may be necessary, analogous to inflating a balloon when small and internal tension is high.\nAfter 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 Rmp, which is orders of magnitude larger than the coil radius Rc (figure not drawn to scale).\nSee RMDCartoon.avi for an animation of this effect.\nThe induced current disc carries a direct current I_{cd} orders of magnitude larger than the input alternating current I_c and forms a static dipole magnetic field oriented perpendicular to the current disc reaching a standoff balance with the plasma wind pressure at distance at the magnetopause boundary according to the [MHD model of an artificial magnetosphere](https://bliptext.com/articles/).\nThe magnetic field falloff rate was assumed in 2001 and 2006 to be f_o =1.\nHowever, as described by Khazanov in 2003 and restated by Slough, Kirtley and Pancotti in 2011 and Kirtley and Slough in a 2012 NIAC report that f_o=2 as demanded by [conservation of magnetic flux](https://bliptext.com/articles/alfv-n-s-theorem).\nSeveral [MPS](https://bliptext.com/articles/) studies concluded that f_o is closer to 2.\nThe falloff rate f_o is a critical parameter in the determination of performance.\nThe RMF-induced rotating disc of electrons has [current density (A m-2)](https://bliptext.com/articles/current-density) j_\\theta(r) at distance r from the antenna for f_o=1 and for f_o=2, which states that flux conservation requires this falloff rate, consistent with a criticism of M2P2 by Cattell as follows:\nwhere B(R_0) T is the magnetic field flux density at radius m near the antenna coils.\nNote that the current density is highest at r=R_0 and falls off at a rate of f_o+1.\nA critical condition for the plasma magnet design provides a lower bound on the RMF frequency 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:\nwhere \\omega_{ci} is the [ion gyrofrequency (s-1)](https://bliptext.com/articles/ion-cyclotron-resonance) in the RMF near the antenna coils, Z is charge number of the ion, e is the [elementary charge](https://bliptext.com/articles/elementary-charge), and m_i kg is the (average) mass of the ion(s).\nSpecifying the magnetic field near the coils at radius R_0 is critical since this is where the current density is greatest.\nChoosing a magnetic field at magnetopause yields a lower value of but ions closer to the coils will rotate.\nAnother condition is that be small enough such that collisions are extremely unlikely.\nThe required power to generate the RMF P_{RMF} is derived by integrating the product of the square of the current density from equation and the resistivity of the plasma \\eta_p from R_0 to R_{mp} with the result as follows:\nwhere \\eta_p is the [Spitzer resistivity (W m)](https://bliptext.com/articles/spitzer-resistivity) of the plasma of ~1.2x10−3 T_e^{-3/2} where T_e is the electron temperature assumed to be 15 eV, the same result for f_o=1 and for f_o=2.\nStarting with the definition of plasma wind force from equation, noting that P_w=F_w u rearranging and recognizing that equation gives the solution for B(R_0), which can be substituted and then using equation for B_{mp} yields the following expression\nwhich when multiplied by C_d/2 with C_SO=1 is the same as for f_o=1 Note that solution for P_{RMF} and R_0 must also satisfy equation, to which the comments following regarding a \"tremendous leverage of power\" do not address.\nNote that a number of the examples cited in assume a magnetopause radius R_{mp} that do not meet the MHD applicability test of equation.\nFrom the [definition of power in physics](https://bliptext.com/articles/power-physics) a constant force is power divided by velocity, the force generated by the plasma magnet (PM) sail is as follows from equation Comparing the above with Equation not the dependence on plasma mass density \\rho is of the same form.\nNote from Equation that as the falloff rate f_o increases that the force derived from the plasma wind decreases, or to maintain the same force P_{RMF} and/or R_0 must increase to maintain the same force F_{PM}.\nEquation gives the mass for each physical coil of radius R_c m. Since the RMF requires alternating current and semiconductors are not efficient at higher frequencies, aluminum was specified with mass density \\delta_c = 2,700 kg/m3.\nEstimates of the coil mass are optimistic by a factor of 4 \\pi since only one coil was sized and the coil circumference was specified as R_c instead of 2 \\pi R_c.\nThe coil [resistance](https://bliptext.com/articles/electrical-resistivity-and-conductivity) is the product of coil material [resistivity (Ω m)](https://bliptext.com/articles/electrical-resistivity-and-conductivity) \\eta_c (e.g., ~3x10−8 Ωm for aluminum) and the coil length 2 \\pi R_c divided by the coil wire cross sectional area where r_c is the radius of the coil wire as follows:\nSome additional power must compensate for resistive loss but it is orders of magnitude less than P_{RMF}.\nThe peak current carried by a coil is specified by the RMF power and coil resistance from the definition of [electrical power in physics](https://bliptext.com/articles/power-physics) as follows:\nThe current induced in the disc by the RMF I_{ic} is the integral of the current density j(r,f_o) from equation on the surface of the disc with inner radius R_c and outer radius R_{MP} with result:\nthe same result for f_o=1.\nLaboratory experiments 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.\nSince the scale of a terrestrial experiment is limited, simulations or a flight trial was recommended.\nSome of these concepts adapted to an ionospheric plasma environment were carried on in the [plasma magnetoshell](https://bliptext.com/articles/) design.\nIn 2022 Freeze, Greason and others published a detailed design for a [plasma magnet](https://bliptext.com/articles/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).\nThis design employed a pair of superconducting coils each with radius R_c of 9 m, an alternating current of I_c of 112 A with and a falloff rate of f_o=1.\nA transit time to Jupiter of 25 days was reported for a 21 kg spacecraft design launched in a 16 U Cubesat format.\nUsing f_o=1 creates very optimistic performance numbers, but since Slough changed this to f_o=2 in 2011 and 2012, the case of f_o=1 is not compared in this article.\nAn example for f_o=2 using solar wind parameters \\rho=8x10−21 kg/m3, u=500 km/s then r_g=72 km and B_{mp}=4x10−8 T with R_{mp}=105 m results in r_g < L where MHD applicability occurs.\nWith a coil radius of R_c=1,000 m yields B_0=4x10−4 T from equation.\nThe required RMF power from equation is 13 kW with a required AC coil current I_c=10 A from equation resulting in an induced current of I_{ic}=2 kA from equation.\nWith C_d=5 the plasma magnet force from equation is 197 N. The magnetic force only for the above parameters is 2.8 N from equation and therefore the plasma magnet thrust gain is 71.\nThe [performance comparison](https://bliptext.com/articles/) section gives and optimistic estimate using constant acceleration for f_o=2 results in a transit time of ~100 days.\n# [Plasma magnet principles of operation | upload.wikimedia.org/wikipedia/commons/c/c0/Plasma///magnet///principles///of///operation.jpg]\n\n## Plasma magnetoshell (PMS)\nA 2011 paper by Slough and others and a 2012 NIAC report by Kirtley investigated use of the plasma magnet technology with 1/r2 magnetic field falloff rate for use in the ionosphere of a planet as a braking mechanism in an approach dubbed Plasma magnetoshell (PMS).\nThe magnetoshell creates drag by ionizing neutral atoms in a planet's ionosphere then magnetically deflecting them.\nA tether attaching the plasma magnet coils to the spacecraft transfers momentum such that orbital insertion occurs.\nAnalytical models, laboratory demonstrations and mission profiles to Neptune and Mars were described.\nIn 2017, Kelly described using a single-coil magnet with 1/r3 magnetic field falloff rate and more experimental results.\nIn 2019 Kelly and Little published simulation results for magnetoshell performance scaling.\nA magnet with radius R_c=1 m was tethered to a spacecraft with batteries for 1,000 seconds of operation (longer than aerocapture designs).\nThe simulations assumed a magnet mass M_c=1,000 kg and total magnetoshell system mass of 1,623 kg, suitable for a [Cassini–Huygens](https://bliptext.com/articles/cassini-huygens) or [Juno](https://bliptext.com/articles/juno-spacecraft) size orbiter.\nThe planet's mass and atmosphere atomic composition and density determine a threshold velocity where magnetoshell operation is feasible.\nSaturn and Neptune have a hydrogen atmosphere and a threshold velocity of approximately 22 km/s.\nIn a Neptune mission a \\Delta v=6 km is required for a 5,000 kg spacecraft and must average 50 kN for the maneuver duration.\nThe model overestimates performance for N2 (Earth, Titan) and CO2 (Venus, Mars) atmospheres since multiple ion species are created and more complex interactions occur.\nFurthermore, the relatively lower mass of Venus and Mars reduces the threshold velocity below that of feasible magnetoshell operation.\nThe paper states that [aerocapture](https://bliptext.com/articles/aerocapture) technologies are mature enough for these mission profiles.\nIn 2021, Kelly and Little published further details for use of drag-modulated plasma aerocapture (DMPA) that when compared to Adaptable Deployable Entry and Placement Technology (ADEPT) for drag-modulated aerocapture (DMA) to Neptune that could deliver 70% higher orbiter mass and experience 30% lower stagnation heating.\n\n## Beam powered magsail (BPM)\nA [beam-powered](https://bliptext.com/articles/beam-powered-propulsion) of M2P2 variant, [MagBeam](https://bliptext.com/articles/magnetized-beamed-plasma-propulsion) was proposed in 2011.\nIn this design a magnetic sail is used with [beam-powered propulsion](https://bliptext.com/articles/beam-powered-propulsion), by using a high-power [particle accelerator](https://bliptext.com/articles/particle-accelerator) to fire a beam of charged particles at the spacecraft.\nThe magsail would deflect this beam, transferring momentum to the vehicle, that could provide higher acceleration than a solar sail driven by a [laser](https://bliptext.com/articles/laser), but a charged particle beam would disperse in a shorter distance than a [laser](https://bliptext.com/articles/laser) due to the electrostatic repulsion of its component particles.\nThis 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](https://bliptext.com/articles/jordin-kare).\n\n# Performance comparison\nThe table below compares [performance measures](https://bliptext.com/articles/) for the magnetic sail designs with the following parameters for the solar wind (sw) at 1 AU: velocity u_{sw}= 500 km/s, number density n_i= 5x106 m−3, ion mass m_i = 1.67x10−27 kg a [proton mass](https://bliptext.com/articles/proton-mass), resulting in mass density = 8.4x10−21 kg/m3, and coefficient of drag C_d=5 where applicable.\nEquation gives the magnetic field at magnetopause as B_{mp}≈ 36 nT, equation gives the ion gyroradius r_g≈ 72 km for C_{Li}=2.\nTable entries in boldface are from a cited source as described in the following:\nEquation determines force for the [Magsail (MS)](https://bliptext.com/articles/) divided by the Freeland correction factor 3.1, equation defines the force for the [plasma magnet (PM)](https://bliptext.com/articles/) with the assumed magnetic field falloff rate f_o=2.\nThe force for the magnetic sail alone F_{mag} is from equation.\nThrust gain G_T for the [magneto plasma sail (MPS)](https://bliptext.com/articles/) is the simulation and/or experimentally determined value with force defined equation to account for thrust loss due to operation in a kinematic region.\nThe last column headed MPS+MPD adds a [magnetoplasma dynamic thruster](https://bliptext.com/articles/magnetoplasmadynamic-thruster) (MPD) that has a higher thrust gain as determined by experiment and simulation.\nFurther details are in the section for the specific design.\nFor designs other than MPS and MPS+MPD, the thrust gain G_T is the achieved force from the first row divided by the force of a magnetic sail alone in the second row.\nThe magnetopause distance and the coil radius R_c are design parameters.\nEquation with r=R_{mp} defines the magnetic field near the coil(s) as.\nThe superconducting coil designs used a critical current density J_c=2x106 A/m to account for warmer temperatures in the solar system.\nThe plasma magnet uses AC power for the rotating magnetic field, PRMF as specified in Equation and cannot use a superconducting coil and assumed an aluminum coil with material density \\delta_c = 2,700 kg/m3 and coil wire radius r_c=5 mm.\nAll other designs assumed a superconducting coil with material density \\delta_c =6,500 kg/m3, coil wire radius r_c=5 mm, and critical current 1.6 x106 A, above which the coil becomes a normal conductor.\nThe magnetopause distance and coil radius R_c for superconducting-coil based designs were adjusted to meet this critical current constraint.\nThe values for the plasma magnet used a value of R_c for f_o=2 selected to minimize time to velocity and distance.\nThe MPS values for and R_c were chosen to match the thrust gain from simulation and scaled experimental results and meet the superconducting-coil critical current constraint.\nEquation gives the physical coil mass M_c(phy) assuming a coil wire radius r_c=5 mm. Equation gives the plasma magnet alternating current I_c.\nEquation gives the direct current I_c with C_0=2 for all other designs.\nThe plasma magnet RMF uses the input alternating current I_c (kA) to rotate electrons in captured plasma to create an induced direct current disc carrying I_{ic} kA as defined in equation.\nSuperconducting coils do not require continuous power (except possibly for cooling); however, the plasma magnet design does, as specified in equation.\nAn estimate for the plasma magnet power supply mass assumes ~3 kg/W for [nuclear power in space](https://bliptext.com/articles/nuclear-power-in-space).\nOther mass was assumed to be 10 tonnes for MS and 1 tonne for PM and MPS. Acceleration a is the thrust force F from the first row divided by the total mass (coil plus other).\nAn optimistic approximation is constant acceleration a, for which the time to reach a target velocity V of 10% of the solar wind velocity is and time to cover a specified distance D ≈ 7.8x108 km (approximate distance from Earth to Jupiter) is.\nFor comparison purposes the time for a [Hohmann transfer](https://bliptext.com/articles/delta-v-budget) from Earth orbit to Jupiter orbit is 2.7 years (almost 1,000 days) but that would allow orbital insertion whereas a magnetic sail would do a flyby unless the magnetosphere and gravity of Jupiter could provide deceleration.\nAnother comparison is the [New Horizons](https://bliptext.com/articles/new-horizons) interplanetary space probe with a 30 kg payload that flew by Jupiter after 405 days on its way to Pluto.\nThe best time to velocity T_V and distance T_D performance occurs for the PM and MPS designs due primarily to much reduced coil and other mass.\nAs described in the [M2P2 section](https://bliptext.com/articles/), several criticisms asserted that the falloff rate f_o=1 was questionable and hence it was not included in this table.\nSimulations and experiments as described in the [MPS section](https://bliptext.com/articles/) showed that f_o=2 is valid with injection of plasma to inflate the magnetic field in a manner similar to M2P2.\nAs described in the [PM section](https://bliptext.com/articles/), plasma is not injected but instead captured to achieve a falloff rate of f_o=2, with calculations assuming f_o=1 being very optimistic.\nThe classic [Magsail (MS)](https://bliptext.com/articles/) design generates the most thrust force and has considerable mass but still has relatively good time performance.\nParameters for the other designs were chosen to yield comparable time performance subject to the constraints previously described.\nAs described above and further detailed in the section for the respective design, this article contains the equations and parameters to compute performance estimates for different parameter choices.\n\n# Criticisms\nIn 1994 Vulpetti published a critical review regarding viability of space propulsion based on solar wind momentum flux.\nThe paper highlighted technology challenges in terms of the magnetic field source, energy required and interaction between the solar wind and the spacecraft's magnetic field, summarizing that these issues were not insurmountable.\nThe major unresolved issue is spacecraft and mission design that account for the potentially highly variable solar wind velocity and plasma density that could complicate maneuvers by a spacecraft employing magnetic sail technology.\nSome means of modulating thrust is necessary.\nIf the mission objective is to rapidly escape the solar system then the paper states that this is less of an issue.\nIn 2006, Bolonkin published a paper that questioned the theoretical viability of a Magsail and described common mistakes.\nEquation (2) states that the magnetic field of electrons rotating in the large coil was greater than and opposed the magnetic field generated by the current in the coil and hence no thrust would result.\nIn 2014 Vulpetti published a rebuttal that summarized plasma properties, in particular the fact that plasma is quasi-neutral and noted in equation (B1) that the Bolonkin paper equation (2) assumed that the plasma had a large net negative electrical charge.\nThe plasma charge varies statistically over short intervals and the maximum value has negligible impact on Magsail performance.\nFurthermore, he argued that observations by many spacecraft have observed compression of a magnetic field by dynamic (or ram) pressure that did not depend on particle charges.\nIn 2017, Gros published results that differed from prior magsail work.\nA major result was the [Magsail kinetic model](https://bliptext.com/articles/) of equation that is a curve fit to numerical analysis of proton trajectories impacted by a large current carrying superconducting coil.\nThe curve fit scaling relation for the effective sail area A(v) was logarithmic cubed with argument with I the loop current, I_G the curve fit parameter, v the ship velocity and c the speed of light.\nThis differed from the power law scaling of prior work.\nThe Gros paper could not trace back this difference to underlying physical arguments and noted that the results are inconsistent, stating that the source for these discrepancies was unclear.\nAppendix B questioned whether a bow shock will form if the initial spacecraft velocity v_0 is large, for example for deceleration after interstellar travel, since the predicted effective sail area A_G(v) is small in this case.\nOne difference is that this analysis used the coil radius R_c for computation of the ion gyroradius as compared with prior work use of the magnetopause radius R_{mp}\n\n# Fictional uses in popular culture\nMagnetic sails have become a popular [trope](https://bliptext.com/articles/trope-literature) in many works of [science fiction](https://bliptext.com/articles/science-fiction) although the [solar sail](https://bliptext.com/articles/solar-sail) is more popular:\nAlthough not referred to as a \"magnetic sail\", the concept was used in the novel [Encounter with Tiber](https://bliptext.com/articles/encounter-with-tiber) by [Buzz Aldrin](https://bliptext.com/articles/buzz-aldrin) and [John Barnes](https://bliptext.com/articles/john-barnes-author) as a braking mechanism to decelerate starships from relativistic speed.","revisionCount":0},"hasMore":false,"nextCursor":null}