Plasma mass density and velocity terminology and units

A plasma consists exclusively of charged particles that can interact with a magnetic or electric field. It does not include neutral particles, such as neutrons. atoms 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 section.

The 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 ${\displaystyle v_{pe}}$. The apparent wind velocity ${\displaystyle u_{sc}}$ as seen by a spacecraft traveling at velocity ${\displaystyle v_{sc}}$ (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 ${\displaystyle u_{sc}=v_{pe}-v_{sc}}$.

Acceleration/deceleration in a stellar plasma wind

Many designs, analyses, simulations and experiments focus on using a magnetic sail in the solar wind plasma to accelerate a spacecraft away from the Sun.^[2] Near the Earth's orbit at 1 AU the plasma flows at velocity ${\displaystyle v_{sw}}$ 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^[25] Assuming that 8% of the solar wind is helium and the remainder hydrogen, the average solar wind plasma mass density at 1 AU is ${\displaystyle 4\times 10^{-21}<\rho _{sw}(1)<10^{-20}}$ kg/m³ (typically 10⁻²⁰ kg/m³).^[26]

The average plasma mass density of ions ${\displaystyle \rho _{sw}}$ decreases according to an Inverse-square law with the distance from the Sun as stated by Andrews/Zubrin^[27] and Borgazzi.^[28] The velocity for values near the Sun is nearly constant, falling off slowly after 1 AU^{[28]: Fig 5} and then rapidly decreases at heliopause.

Deceleration in interstellar medium (ISM)

A 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. As shown in the section on Magsail kinematic model (MKM), feasible uses of this involve maximum velocities below 10% c, taking decades to decelerate, for total travel times on the order of a century as described in the magsail specific designs section.

Only the magsail references consider deceleration in the ISM on approach to Alpha (${\displaystyle \alpha }$) Centauri, which as shown in the figure is separated by the local bubble and the G-clouds and the Solar System, which is moving at velocity ${\displaystyle v_{sun}}$and the local cloud is moving at velocity ${\displaystyle v_{L|C}}$. Estimates of the number of protons range between 0.005 and 0.5 cm⁻³ resulting in a plasma mass density ${\displaystyle 9\times 10^{-24}<\rho _{im}<3\times 10^{-22}}$ kg/m³ , which covers the range used by references in the magsail specific designs section. As 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. A typical value assumed for approach to Alpha Centauri is a proton number density ${\displaystyle n_{i}}$of 0.1 protons per cm^3[29] corresponding to ${\displaystyle \rho _{im}\approx 10^{-22}}$ kg/m³.

The spacecraft velocity ${\displaystyle v_{sc}}$ 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 ${\displaystyle u_{im}\approx -v_{sc}}$.

Radio emissions of 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 ${\displaystyle 120\,v_{sc}/c}$ kHz.^[30] The 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. Detection of such radiation could indicate activity of advanced extraterrestrial civilizations.

In a planetary ionosphere

A 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.^[18][20] The plasma mass in a planetary ionosphere (pi) ${\displaystyle \rho _{pi}}$ is composed of multiple ion types and varies by altitude. The spacecraft velocity ${\displaystyle v_{sc}}$ is much greater than the planetary ionosphere velocity in a deceleration maneuver so the apparent plasma wind velocity is approximately ${\displaystyle u_{pi}\approx -v_{sc}}$ at the beginning of a deceleration maneuver.

In a planetary magnetosphere

Inside or near a planetary magnetosphere, a magnetic sail can thrust against or be attracted to a planet's magnetic field created by a dynamo, especially in an orbit that passes over the planet's magnetic poles.^[5] When 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.

The thrust that a magnetic sail delivers within a magnetosphere decreases with the fourth power of its distance from the planet's internal magnetic field. When close to a planet with a strong magnetosphere such as Earth or a gas giant, the magnetic sail could generate more thrust by interacting with the magnetosphere instead of the solar wind. When 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.

By varying the magnetic sail's field strength and orientation a "perigee kick" can be achieved raising the altitude of the orbit's apogee higher and higher, until the magnetic sail is able to leave the planetary magnetosphere and catch the solar wind. The 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.

In 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. However, this requires the magnetic sail to be maintained in an "unstable" orientation. Furthermore, 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.^[5]

In 2022 a spaceflight trial dubbed Jupiter Observing Velocity Experiment (JOVE) proposed using a plasma magnet to decelerate against the magnetosphere of Jupiter.^[17]

Magnetic field interaction with charged particles

An ion or electron with charge q in a plasma moving at velocity v in a magnetic field B and electric field E is treated as an idealized point charge in the Lorentz force ${\displaystyle \mathbf {F} =q\,\mathbf {E} +q\,\mathbf {v} \times \mathbf {B} }$. This means that the force on an ion or electron is proportional to the product of their charge q and velocity component ${\displaystyle v}$ perpendicular to the magnetic field flux density B, in SI units as teslas (T). A 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.^[2] An electric sail uses an electric field E that under certain conditions interact with charged particles to create thrust.

Artificial magnetospheric model

Artificial magnetospheric model

The characteristics of the Earth's magnetosphere have been widely studied as a basis for magnetic sails. The 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. A source attached to a spacecraft generates a magnetic field. Under certain conditions at the boundary where magnetic pressure equals the plasma wind kinetic pressure an artificial bow shock and magnetopause forms at a characteristic length ${\displaystyle L}$ from the field source. The ionized plasma wind particles create a 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.^[1] The magnetopause deflects charged particles, which affects their streamlines and increases the density at magnetopause. A magnetospheric bubble or cavity forms that has very low density downstream from the magnetopause. Upstream from the magnetopause a bow shock develops. Simulation results often show the particle density through use of color with an example shown in the legend in the lower left. This figure uses aspects of the general structure from Zubrin,^{[4]: Fig 3} Toivanen^{[31]: Fig 1} and Funaki^{[1]: Fig 2a} and aspects of the plasma density from Khazanov^{[32]: Fig 1} and Cruz.^{[33]: Fig 2}

Magnetohydrodynamic model

Magnetic sail designs operating in a plasma wind share a theoretical foundation based upon a magnetohydrodynamic (MHD) model, sometimes called a fluid model, from plasma physics for an artificially generated magnetosphere. Under certain conditions, the plasma wind and the magnetic sail are separated by a magnetopause that blocks the charged particles, which creates a drag force that transfers (at least some) momentum to the magnetic sail, which then applies thrust to the attached spacecraft as described in Andrews/Zubrin,^[27] Cattell,^[34]Funaki,^[1] and Toivanen.^[31]

A plasma environment has fundamental parameters, and if a cited reference uses cgs units these should be converted to SI units as defined in the NRL plasma formulary,^[35] which this article uses as a reference for plasma parameter units not defined in SI units. The major parameters for plasma mass density are: the number of ions of type ${\displaystyle i}$ per unit volume ${\displaystyle n_{i}}$ the mass of each ion type accounting for isotopes ${\displaystyle m_{i}}$ and the number of electrons ${\displaystyle n_{e}}$ per unit volume each with electron mass ${\displaystyle m_{e}}$.^[36] An average plasma mass density per unit volume for charged particles in a plasma environment ${\displaystyle pe}$ (${\displaystyle sw}$ for stellar wind, ${\displaystyle pi}$ for planetary ionosphere, ${\displaystyle im}$ for interstellar medium) is expressed in equation form from magnetohydrodynamics as${\displaystyle \textstyle \rho _{pe}=n_{e}m_{e}+\sum _{i}n_{i}m_{i}}$. Note that this definition includes the mass of neutrons in an ion's nucleus. In SI Units per unit volume is cubic metre (m^-3), mass is kilogram (kg), and mass density is kilogram per cubic metre (kg/m³).

Artificial Magnetosphere Model of Basic Magnetic Sail

The figure depicts the MHD model as described in Funaki^[1] and Djojodihardjo.^[2] Starting from the left a plasma wind in a plasma environment (e.g., stellar, ISM or an ionosphere) of effective velocity ${\textstyle v_{pe}}$ with density ${\textstyle \rho _{pe}}$ encounters a spacecraft with time-varying velocity ${\textstyle v_{sc}}$ that is positive if accelerating and negative if decelerating. The apparent plasma wind velocity from the spacecraft's viewpoint is ${\textstyle u_{pe}=v_{pe}-v_{sc}}$. The spacecraft and field source generate a magnetic field that creates a magnetospheric bubble extending out to a magnetopause preceded by a bow shock that deflects electrons and ions from the plasma wind At magnetopause the field source magnetic pressure equals the kinetic pressure of the plasma wind at a standoff shown at the bottom of the figure. The characteristic length ${\textstyle L}$ is that of a circular sail of effective blocking area ${\displaystyle S=\pi \,R_{mp}^{2}}$ where ${\displaystyle R_{mp}\approx L}$ is the effective magnetopause radius. Under certain conditions the plasma wind pushing on the artificial magnetosphere bow shock and magnetopause creates a force ${\textstyle F_{w}}$ on the magnetic field source that is physically attached to the spacecraft so that at least part of the force ${\textstyle F_{w}}$ causes a force ${\textstyle F_{sc}}$on the spacecraft, accelerating it when sailing downwind or decelerating when sailing into a headwind. Under certain conditions and in some designs, some of the plasma wind force may be lost as indicated by ${\textstyle F_{loss}}$ on the right side.

All magnetic sail designs assume a standoff between plasma wind pressure ${\displaystyle p_{w}}$ and magnetic pressure ${\textstyle p_{B}}$ with SI units of Pascal (Pa, or N/m²) differing only in a constant coefficient ${\displaystyle C_{SO}}$ as follows:

${\displaystyle p_{w}=\rho u^{2}=p_{B}={\frac {B_{mp}^{2}}{C_{SO}\mu _{0}}}}$

(MHD.1)

where ${\textstyle u}$ is the apparent wind velocity and ${\displaystyle \rho }$ is the plasma mass density for a specific plasma environment, ${\displaystyle B_{mp}}$ the magnetic field flux density at magnetopause, μ₀ is the vacuum permeability (N A^-2) and ${\textstyle C_{SO}}$ is a constant that differs by reference as follows for ${\textstyle C_{SO}=2}$ corresponding to ${\textstyle p_{w}}$modeled as dynamic pressure with no magnetic field compression,^[31] ${\textstyle C_{SO}=1}$ for ${\textstyle p_{w}}$modeled as ram pressure with no magnetic field compression^[4][16] and ${\textstyle C_{SO}=1/2}$ for ${\textstyle p_{w}}$ modeled as ram pressure with magnetic field compression by a factor of 2^[1] Equation MHD.1 can be solved to yield the required magnetic field ${\displaystyle B_{mp}}$ that satisfies the pressure balance at magnetopause standoff as:

$${\displaystyle B_{mp}=u{\sqrt {\rho \,\mu _{0}\,C_{SO}}}}$$

(MHD.2)

The force with SI Units of Newtons (N) derived by a magnetic sail for a plasma environment is determined from MHD equations as reported by principal researchers Funaki,^[1] Slough,^[16]Andrews and Zubrin,^[27] and Toivanen^[31] as follows:

$${\displaystyle F_{w}=C_{d}\ \rho {\frac {u^{2}}{2}}S=C_{d}\ \rho {\frac {u^{2}}{2}}\pi L^{2}}$$

(MHD.3)

where ${\textstyle C_{d}}$ is a coefficient of drag determined by numerical analysis and/or simulation, ${\textstyle \rho u^{2}/2}$ is the wind pressure, and ${\displaystyle S=\pi R_{mp}^{2}}$ is the effective blocking area of the magnetic sail with magnetopause radius ${\textstyle R_{mp}\approx L}$. Note that this equation has the same form as the drag equation in fluid dynamics. ${\textstyle C_{d}}$ is a function of coil attack angle on thrust and steering angle. The power (W) of the plasma wind is the product of velocity and a constant force

$${\displaystyle P_{w}=u\,F_{w}=C_{d}\ \rho {\frac {u^{3}}{2}}\pi R_{mp}^{2}={\frac {C_{d}u\pi }{2\mu _{0}C_{SO}}}R_{mp}^{2}B_{mp}^{2}}$$

(MHD.4)

where equation MHD.2 was used to derive the right side.^{[16]: Eq (9)}

MHD applicability test

As summarized in the overview section, an important condition for a magnetic sail to generate maximum force is that the magnetopause radius be on the order of an ion's radius of gyration. Through analysis, numerical calculation, simulation and experimentation an important condition for a magnetic sail to generate significant force is the MHD applicability test,^[37] which states that the standoff distance ${\displaystyle L}$ must be significantly greater than the ion gyroradius, also called the Larmor radius^[1] or cyclotron radius:

$${\displaystyle r_{g}={\frac {m_{i}v_{\perp }}{\mid q\mid B_{x}C_{Li}}}}$$

(MHD.5)

where ${\textstyle m_{i}}$ is the ion mass, ${\displaystyle v_{\perp }}$ is the velocity of a particle perpendicular to the magnetic field, ${\displaystyle |q|}$ is the elementary charge of the ion, ${\textstyle B_{x}}$ is the magnetic field flux density at the point of reference ${\textstyle x}$ and ${\textstyle C_{Li}}$ is a constant that differs by source with ${\textstyle C_{Li}=1}$^[16] and ${\textstyle C_{Li}=2}$^[1]. For example, in the solar wind with 5 ions/cm³ at 1 AU with ${\displaystyle m_{i}}$ the proton mass (kg) , ${\displaystyle v_{\perp }=v_{sw}}$ = 400 km/s, ${\displaystyle B_{x}=B_{mp}}$ = 36 nT with ${\displaystyle C_{SO}}$=0.5 from equation MHD.2 at magnetopause and ${\textstyle C_{Li}}$=2 then ${\displaystyle r_{g}\approx }$ 72 km.^{[1]: Eq (7)} The MHD applicability test is the ratio ${\textstyle r_{g}/L}$. The figure plots ${\displaystyle C_{d}}$ on the left vertical axis and lost thrust on the right vertical axis versus the ratio ${\textstyle r_{g}/L}$. When ${\displaystyle r_{g}/L<1}$, ${\displaystyle C_{d}=3.6}$ is maximum, at ${\textstyle r_{g}/L\approx 1}$, ${\textstyle C_{d}=2.7}$, a decrease of 25% from the maximum and at ${\textstyle r_{g}/L\approx 2}$, ${\textstyle C_{d}\approx 1.5}$, a 45% decrease. As ${\displaystyle r_{g}/L}$ increases beyond one, ${\displaystyle C_{d}}$ decreases meaning less thrust from the plasma wind transfers to the spacecraft and is instead lost to the plasma wind. In 2004, Fujita^[38][1] 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 ${\displaystyle C_{d}}$ for a magnetic sail operating in the radial orientation resulting in the following approximate formula:

$${\displaystyle C_{d}(r_{g}/L)={\begin{cases}3.6\,e^{-0.28(r_{g}/L)^{2}},&{\text{for }}r_{g}/L<1\\{\frac {3.4}{r_{g}/L}}e^{-0.22(L/r_{g})^{2}},&{\text{for }}r_{g}/L\geq 1\end{cases}}}$$

(MHD.6)

The lost thrust is ${\displaystyle T_{loss}=(1-C_{d}(r_{g}/L))/3.6}$.

Coil attack angle effect on thrust and steering angle

Coil magnetic field orientation and forces

In 2005 Nishida and others published results from numerical analysis of an MHD model for interaction of the solar wind with a magnetic field of current flowing in a coil that momentum is indeed transferred to the magnetic field produced by field source and hence to the spacecraft.^[39] Thrust force derives from the momentum change of the solar wind, pressure by the solar wind on the magnetopause from equation MHD.1 and Lorentz force from currents induced in the magnetosphere interacting with the field source. The results quantified the coefficient of drag, steering (i.e., thrust direction) angle with the solar wind, and torque generated as a function of attack angle (i.e., orientation) The figure illustrates how the attack (or coil tilt) angle ${\displaystyle \alpha _{t}}$ orientation of the coil creates a steering angle for the thrust vector and also torque imparted to the coil. Also shown is the vector for the interplanetary magnetic field (IMF), which at 1 AU varies with waves and other disturbances in the solar wind, known as space weather, and can significantly increase or decrease the thrust of a magnetic sail.^[40]

For a coil with radial orientation (like a Frisbee) the attack angle ${\displaystyle \alpha _{t}}$= 0° and with axial orientation (like a parachute) ${\displaystyle \alpha _{t}}$=90°. The Nishida 2005 results^[39] reported a coefficient of drag ${\displaystyle C_{d}}$ that increased non-linearly with attack angle from a minimum of 3.6 at ${\displaystyle \alpha _{t}}$=0 to a maximum of 5 at ${\displaystyle \alpha _{t}}$=90°. The steering angle of the thrust vector is substantially less than the attack angle deviation from 45° due to the interaction of the magnetic field with the solar wind. Torque increases from ${\displaystyle \alpha _{t}}$= 0° from zero at to a maximum at ${\displaystyle \alpha _{t}}$=45° and then decreases to zero at ${\displaystyle \alpha _{t}}$=90°. A number of magnetic sail design and other papers cite these results. In 2012 Kajimura reported simulation results^[41] that covered two cases where MHD applicability occurs with ${\displaystyle r_{g}/L}$=1.125 and where a kinematic model is applicable ${\displaystyle r_{g}/L}$=0.125 to compute a coefficient of drag ${\displaystyle C_{d}}$ and steering angle. As shown in Figure 4 of that paper when MHD applicability occurs the results are similar in form to Nishida 2005^[39] where the largest ${\displaystyle C_{d}}$ occurs with the coil in an axial orientation. However, when the kinematic model applies, the largest ${\displaystyle C_{d}}$ occurs with the coil in an radial orientation. The steering angle is positive when MHD is applicable and negative when a kinematic model applies. The 2012 Nishida and Funaki published simulation results^[42] for a coefficient of drag ${\displaystyle C_{D}}$, coefficient of lift ${\displaystyle C_{L}}$ and a coefficient of moment ${\displaystyle C_{M}}$ for a coil radius of ${\displaystyle R_{c}}$=100 km and magnetopause radius ${\displaystyle R_{mp}}$=500 km at 1 AU.

Magnetic field model

In a design, either the magnetic field source strength or the magnetopause radius ${\displaystyle R_{mp}\approx L}$ the characteristic length must be chosen. A good approximation from Cattell^[34] and Toivanen^[31] for a magnetic field falloff rate ${\displaystyle f_{o},1\leq f_{o}\leq 3}$ for a distance ${\textstyle R_{0}\leq r\leq R_{mp}}$ from the field source to magnetopause starts with the equation:

${\displaystyle B(r)\approx B(R_{0}){\biggl (}{\frac {R_{0}}{r}}{\biggr )}^{f_{o}}}$

(MFM.1)

where ${\displaystyle B(R_{0})}$ is the magnetic field at a radius ${\displaystyle R_{0}}$ near the field source that falls off near the source as ${\displaystyle 1/r^{3}}$ as follows:

${\displaystyle B(R_{0})\approx C_{0}{\frac {\mu _{0}\,\mathbf {m} }{4\,\pi \,R_{0}^{3}}}}$

(MFM.2)

where ${\displaystyle C_{0}}$ is a constant multiplying the magnetic moment (A m²) ${\displaystyle \mathbf {m} }$ to make ${\displaystyle B(r)}$ match a target value at ${\textstyle r>>R_{0}}$. When far from the field source, a magnetic dipole is a good approximation and choosing the above value of ${\displaystyle B(R_{0})}$ with ${\displaystyle C_{0}}$ =2 near the field source was used by Andrews and Zubrin.^[4]

The Amperian loop model for the magnetic moment is ${\displaystyle \mathbf {m} =I_{c}S}$, where ${\displaystyle I_{c}}$ is the current in amperes (A) and ${\displaystyle S=\pi \,R_{c}^{2}}$ is the surface area for a coil (loop) of radius ${\displaystyle R_{c}}$. Assuming that ${\displaystyle R_{0}\approx R_{c}}$ and substituting the expression for the magnetic moment ${\displaystyle \mathbf {m} }$ into equation MFM.2 yields the following:

${\displaystyle I_{c}={\frac {4}{C_{0}\mu _{0}}}B(R_{c})\,R_{c}}$

(MFM.3)

When the magnetic field flux density ${\displaystyle B(R_{0})}$ is specified, substituting ${\displaystyle B_{mp}}$ from the pressure balance analysis from equation MHD.2 into the above and solving for ${\textstyle R_{mp}}$ yields the following:

${\displaystyle L\approx R_{mp}\approx R_{0}{\biggl (}{\frac {B(R_{0})}{B_{mp}}}{\biggr )}^{1/f_{o}}}$

(MFM.4)

This is the expression for ${\displaystyle L}$ when ${\displaystyle f_{o}=3}$ with ${\displaystyle C_{SO}=1/2}$^{[1]: Eq (4)} and ${\displaystyle C_{SO}=2}$^{[31]: Eq (4)} and is the same form as the magnetopause distance of the Earth. Equation MFM.4 shows directly how a decreased falloff rate ${\displaystyle f_{o}}$ dramatically increases the effective sail area ${\displaystyle S=\pi R_{mp}^{2}}$ for a given field source magnetic moment ${\displaystyle \mathbf {m} }$ and ${\displaystyle B_{mp}}$ determined from the pressure balance equation MHD.1. Substituting this into equation MHD.3 yields the plasma wind force as a function of falloff rate ${\displaystyle f_{o}}$, plasma density ${\displaystyle \rho }$, coil radius ${\displaystyle R_{c}\approx R_{0}}$, coil current ${\displaystyle I_{c}}$ and plasma wind velocity ${\displaystyle u}$ as follows:

${\displaystyle F_{w}(f_{o})={\frac {C_{d}}{2}}\rho u^{2}\,\pi \,R_{0}^{2}\,{\biggl (}{\frac {B(R_{0})}{B_{mp}}}{\biggr )}^{2/f_{o}}={\frac {C_{d}}{2}}\rho u^{2}\,\pi \,R_{0}^{2}\,{\biggl (}{\frac {{\sqrt {\mu _{0}}}I_{c}C_{0}}{4R_{0}\,u{\sqrt {\rho \,C_{SO}}}}}{\biggr )}^{2/f_{o}}}$

(MFM.5)

using equation MFM.3 for ${\displaystyle B(R_{0})}$ and equation MHD.2 for ${\displaystyle B_{mp}}$. This is the same expression as equation (10b) when ${\displaystyle f_{o}=3}$ and ${\displaystyle C_{SO}=1/2}$^[2] and^{[7]: Eq (108)} and the right hand side from equation (20) specifically applied to M2P2^[31] with other numerical coefficients grouped into the ${\displaystyle C_{d}}$ term. Note that force increases as falloff rate decreases. For the solar wind case, substituting MHD.2 into MFM.5 and using the function for the solar wind plasma mass density ${\displaystyle \rho _{sw}(a_{\odot })=\rho _{sw}(1)/a_{\odot }^{2}}$,^{[28]: Fig 5} with ${\displaystyle a_{\odot }}$ the distance from the sun in Astronomical units (AU) results in the following expression:

${\displaystyle F_{sw}(f_{o},a_{\odot })=C_{f_{o}}\,(\rho _{sw}(a_{\odot })\,u^{2}))^{1-1/f_{o}}\,R_{0}^{2}\,S}$

(MFM.6)

where ${\displaystyle C_{f_{o}}={\frac {C_{d}}{2}}{\biggl (}{\frac {B(R_{0})^{2}}{\mu _{0}C_{SO}}}{\biggr )}^{1/f_{o}},{\mathsf {and}}\,S=\pi \,R_{mp}^{2}}$, the effective sail blocking area.

This equation explicitly shows the relationship upon solar wind plasma mass density ${\displaystyle \rho _{sw}(a_{\odot })}$ as a function of distance from the Sun ${\displaystyle a_{\odot }}$. For the case ${\displaystyle f_{o}}$=1 the expansion of the magnetopause radius ${\displaystyle R_{mp}}$ exactly matches the decreasing value of ${\displaystyle \rho _{sw}(a_{\odot })}$ exactly as the distance from the Sun ${\displaystyle a_{\odot }}$ increases, resulting in constant force and hence constant acceleration inside the heliosphere.^[16] Note that ${\displaystyle C_{f_{o}}}$ includes the term ${\displaystyle B(R_{0})^{2/f_{o}}}$, which means that as ${\displaystyle f_{o}}$ increases that the magnetic field near the field source ${\displaystyle B(R_{0})}$ must increase to maintain the same force as compared with a smaller value of ${\displaystyle f_{o}}$. The example in the overview section set ${\displaystyle C_{f_{o}}}$=1, ${\displaystyle u}$=1, ${\displaystyle R_{0}}$=1, and ${\displaystyle S}$=1 so that the force at ${\displaystyle a_{\odot }}$=1 was equal to 1 for all values of ${\displaystyle f_{o}}$ at 1 AU.

General kinematic model

When the MHD applicability test of ${\textstyle r_{g}/L}$ <1 then a kinematic simulation model more accurately predicts force transferred from the plasma wind to the spacecraft. In this case the effective sail blocking area ${\displaystyle A_{e}}$ < ${\displaystyle S=\pi L^{2}}$.

Magnetic dipole force: MHD and kinematic models

The left axis of the figure is for plots of magnetic sail force versus characteristic length ${\displaystyle L}$. The solid black line plots the MHD model force ${\displaystyle F_{MHD}}$ from equation MHD.3. The green line shows the value of ion gyroradius ${\displaystyle r_{g}\approx }$ 72 km at 1 AU from equation MHD.5. The dashed blue line plots the hybrid MHD/kinematic model from equation MHD.6 from Fujita04.^[38] The red dashed line plots a curve fit to simulation results from Ashida14.^[24] Although a good fit for these parameters, the curve fit range of this model does not cover some relevant examples. Additional simulation results from Hajiwara15^[43] are shown for the MHD and kinematic model as single data points as indicated in the legend. These models are all in close agreement. The kinematic models predict less force than predicted by the MHD model. In other words, the fraction ${\displaystyle T_{loss}}$ of thrust force predicted by the MHD model is lost when ${\displaystyle r_{g}/L<1}$ as plotted on the right axis. The solid blue and red lines show ${\displaystyle T_{loss}}$ for Fujita04^[38] and Ashida18^[24] respectively, indicating that operation with ${\displaystyle L}$ less than 10% of ${\displaystyle r_{g}}$ will have significant loss. Other factors in a specific magnetic sail design may offset this loss for values of ${\displaystyle L<r_{g}}$.

Performance measures

Important measures that determine the relative performance of different magnetic sail systems include: mass of the field source generator and its power and energy requirements; thrust achieved; thrust to weight ratio, any limitations and constraints, and propellant system exhausted, if any . Mass of the field source ${\displaystyle M_{fs}}$ in the Magsail design was relatively large and subsequent designs strove to reduce this measure. Total spacecraft mass is ${\displaystyle M_{sc}=M_{fs}+M_{p}}$, where ${\displaystyle M_{p}}$ is the payload mass. Power requirements are significant in some designs and add to field source mass. Thrust is the plasma wind force ${\displaystyle F_{w}}$ for a particular plasma environment with acceleration ${\displaystyle a=F_{w}/M_{sc}}$. The thrust to weight ratio ${\textstyle F_{w}/M_{sc}}$is also an important performance measure. Other limitations and constraints may be specific to a particular design. The M2P2 and MPS designs, as well as potentially the plasma magnet design, exhaust some plasma as part of inflating the magnetospheric bubble and these cases also have a specific impulse and effective exhaust velocity performance measure.

Magsail (MS)

Andrews was working on use of a magnetic scoop to gather interstellar material as propellant for a nuclear electric ion drive spacecraft, allowing the craft to operate in a similar manner to a Bussard ramjet, whose history goes back to at least 1973.^[44] Andrews asked Zubrin to help compute the magnetic scoop drag against the interplanetary medium, which turned out to be much greater than the ion drive thrust. The ion drive component of the system was dropped, and use of the concept of using the magnetic scoop as a magnetic sail or Magsail (MS) was born.^[45]

Andrews & Zubrin Magsail

The figure shows the magsail design^[4] consisting of a loop of superconducting wire of radius ${\displaystyle R_{c}}$ on the order of 100 km that carries a direct current ${\displaystyle I_{c}}$ that generates a magnetic field, which was modeled according to the Biot–Savart law inside the loop and as a magnetic dipole far outside the loop. With respect to the plasma wind direction a magsail may have a radial (or normal) orientation or an axial orientation that can be adjusted to provide torque for steering. In non-axial configurations lift is generated that can change the spacecraft's momentum. The loop connects via shroud lines (or tethers) to the spacecraft in the center. Because a loop carrying current is forced outwards towards a circular shape by its magnetic field, the sail could be deployed by unspooling the conductor wire and applying a current through it via the peripheral platforms.^[6] The loop must be adequately attached to the spacecraft in order to transfer momentum from the plasma wind and would pull the spacecraft behind it as shown in the axial configuration in the right side of the figure. This design has a significant advantage of requiring no propellant and is thus a form of field propulsion that can operate indefinitely.^{[27]: Sec VIII}

Magsail kinematic model (MKM)

The MHD applicability test of equation MHD.5 fails in some ISM deceleration cases and a kinematic model is necessary, such as the one documented in 2017 by Claudius Gros summarized here.^[22] A spacecraft with an overall mass ${\displaystyle m_{tot}}$ and velocity ${\displaystyle v}$ follows^{[22]: Eq (1)} of motion as:

${\displaystyle F_{MKM}=m_{tot}{\dot {v}}=-\left(A_{G}(v)n_{p}v\right)(2m_{p}v)=2\,\rho _{im}\,v^{2}A_{G}(v),}$

(MKM.1)

where ${\displaystyle F_{MKM}}$ N is force predicted by this model, ${\displaystyle n_{p}}$ m⁻³ is the proton number density, ${\displaystyle m_{p}}$ kg is the proton mass, ${\displaystyle \rho =m_{p}n_{p}}$ kg/m³ the plasma density, and ${\displaystyle A_{G}(v)}$ m² the effective reflection area. This equation assumes that the spacecraft encounters ${\displaystyle A_{G}(v)n_{p}v}$ particles per second and that every particle of mass ${\displaystyle m_{p}}$ is completed reflected. Note that this equation is of the same form as MFM.5 with ${\displaystyle C_{d}}$=4, interpreting the ${\displaystyle C_{d}}$ term as just a number.

Gros numerically determined the effective reflection area ${\textstyle A_{G}(v)}$ by integrating the degree of reflection of approaching protons interacting with the superconducting loop magnetic field according to the Biot-Savart law. The reported result was independent of the loop radius ${\displaystyle R_{c}}$. An accurate curve fit as reported in Figure 4 to the numerical evaluation for the effective reflection area for a magnetic sail in the axial configuration from equation (8) was

${\displaystyle A_{G}(v)=0.081\pi R_{c}^{2}\left[\log \left({\frac {cI}{vI_{G}}}\right)\right]^{3}\,\,,v/c<I/I_{G}}$

(MKM.2)

where ${\displaystyle \pi R_{c}^{2}}$ is the area enclosed by the current carrying loop, ${\displaystyle c}$ the speed of light, and the value ${\displaystyle I_{G}=1.55\cdot 10^{6}}$A determined a good curve fit for ${\displaystyle I}$=10⁵ A, the current through the loop. In 2020, Perakis published an analysis^[49] that corroborated the above formula with parameters selected for the solar wind and reported a force no more than 9% less than the Gros model for ${\displaystyle I}$=10⁵ A and ${\displaystyle R_{c}}$=100 m with the coil in an axial orientation.. That analysis also reported on the effect of magsail tilt angle on lift and side forces for a use case in maneuvering within the solar system.

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

${\displaystyle A_{Z}(v)={\frac {1.124\times 1.26}{3.1}}\,L_{Z}}$

(MKM.3)

Magsail MHD and kinematic model effective sail area

The figure shows the normalized effective sail area normalized by the coil area ${\displaystyle \pi R_{c}^{2}}$ for the MKM case from Gros of equation MKM.1 and for Zubrin from equation MKM.3 for ${\displaystyle I\approx I_{G}}$, ${\displaystyle R_{c}}$=100 km, and ${\displaystyle n_{p}}$=0.1 cm⁻³ for the G-cloud on approach to Alpha Centauri corresponding to ISM density ${\displaystyle \rho _{im}=1.67\times 10^{-22}}$ kg/m³ consistent with that from Freeland^[7] plotted versus the spacecraft velocity relative to the speed of light ${\textstyle \beta =v/c}$. A good fit occurs for these parameters, but for different values of ${\displaystyle R_{c}}$ and ${\displaystyle I}$ the fit can vary significantly. Also plotted is the MHD applicability test of ion gyroradius divided by magnetopause radius ${\displaystyle r_{g}/R_{mp}}$ <1 from equation MHD.4 on the secondary axis. Note that MHD applicability occurs at ${\displaystyle v/c}$ < 1%. For comparison, the 2004 Fujita ${\displaystyle C_{d}}$ as a function of ${\displaystyle r_{g}/R_{mp}}$ from the MHD applicability test section is also plotted. Note that the Gros model predicts a more rapid decrease in effective area than this model at higher velocities. The normalized values of ${\displaystyle A_{G}(v)}$ and ${\displaystyle A_{Z}(v)}$ track closely until ${\displaystyle \beta =v/c\approx }$ 10% after which point the Zubrin magsail model of Equation MS.4 becomes increasingly optimistic and equation MKM.2 is applicable instead. Since the models track closely up to ${\displaystyle \beta \approx }$ 10%, with the kinematic model underestimating effective sail area for smaller values of ${\displaystyle \beta }$ (hence underestimating force), equation MKM.1 is an approximation for both the MHD and kinematic region. The Gros model is pessimistic for ${\displaystyle \beta }$ < 0.1%.

Gros used the analytic expression for the effective reflection area ${\displaystyle A_{G}(v)}$ from equation MKM.3 for explicit solution for the required distance ${\displaystyle x_{f}}$ m to decelerate to final velocity ${\displaystyle v_{f}\approx 0.013c}$ m/s from^{[22]: Eq (10)} given an initial velocity ${\displaystyle v_{0}}$ m/s for a spacecraft mass ${\displaystyle m_{tot}}$ kg as follows:

${\displaystyle x_{f}={\frac {m_{tot}\,[g(v_{0})-g(v_{f})]}{0,081\,m_{p}\,n_{p}\pi \,R_{c}^{2}}}}$

(MKM.4)

where ${\textstyle g(v)=ln^{-2}({\frac {v\,I_{G}}{c\,I}})}$. When ${\displaystyle v_{f}}$=0 the above equation is defined in^{[22]: Eq (11)} as ${\displaystyle x_{max}}$, which enabled a closed form solution of the velocity at a distance ${\displaystyle x,v(x)}$ in^{[22]: Eq (12)} with numerical integration required to compute the time required to decelerate.^{[22]: Eq (14)} Equation (16) The optimal current that minimized ${\displaystyle x_{max}}$ as ${\displaystyle I_{opt}=e\beta _{0}I_{G}}$ where ${\displaystyle \beta _{0}=v_{0}/c}$.^{[22]: Eq (16)} In 2017 Crowl^[47] optimized coil current for the ratio of effective area ${\displaystyle A(v)}$ over total mass ${\displaystyle m_{tot}}$ and derived the result ${\displaystyle I_{opt}=e^{3}\,\beta _{0}I_{G}}$.^{[22]: Eq (15)} That paper used results from Gros for the stopping distance ${\displaystyle x_{max}}$ and time to decelerate.

Magsail ISM deceleration distance and time comparison

The figure plots the distance traveled while decelerating ${\displaystyle x_{d}}$ and time required to decelerate ${\displaystyle t_{d}}$ given a starting relative velocity ${\displaystyle \beta _{0}=v_{0}/c}$ and a final velocity ${\displaystyle v_{f}=0.013c}$ m/s consistent with that from Freeland^[7] for the same parameters above. Equation CMC.1 gives the magsail mass ${\displaystyle M_{s}}$ as 97 tonnes assuming payload mass ${\displaystyle M_{p}}$ of 100 tonnes using the same values used by Freeland^[7] of ${\displaystyle J_{e}}$ = 10¹¹ A/m² and ${\displaystyle \delta _{c}}$=6,500 kg/m³ for the superconducting coil. Equation MS.4 gives Force for the magsail multiplied by ${\displaystyle C_{d}}$=4 for the Andrews/Zubrin model to align with equation MHD.3 definition of force from the Gros model. Acceleration is force divided by mass, velocity is the integral of acceleration over the deceleration time interval ${\displaystyle t_{d}}$ and deceleration distance traveled ${\displaystyle x_{d}}$ is the integral of the velocity over ${\displaystyle t_{d}}$. Numerical integration resulted in the lines plotted in the figure with deceleration distance traveled plotted on the primary vertical axis on the left and time required to decelerate on the secondary vertical axis on the right. Note that the MHD Zubrin model and the Gros kinematic model predict nearly identical values of deceleration distance up to ${\displaystyle \beta _{0}}$~ 5% of c, with the Zubrin model predicting less deceleration distance and shorter deceleration time at greater values of ${\displaystyle \beta _{0}}$. This is consistent with the Gros model predicting a smaller effective area ${\displaystyle A(v)}$ at larger values of ${\displaystyle \beta _{0}}$. The value of the closed form solution for deceleration distance ${\displaystyle x_{f}}$from MKM.4 for the same parameters closely tracks the numerical integration result.

Mini-magnetospheric plasma propulsion (M2P2)

Winglee M2P2 schematic

In 2000 Winglee, Slough and others proposed a design order to reduce the size and weight of a magnetic sail well below that of the magsail and named it mini-magnetospheric plasma propulsion (M2P2) that reported results adapted from a simulation model of the Earth's magnetosphere.^[8] The figure based upon Winglee,^[8] Hajiwara,^[56] Arita,^[57] and Funaki^[10] illustrates the M2P2 design, which was the basis of the subsequent Magneto plasma sail (MPS) design. Starting at the center with a solenoid coil of radius ${\displaystyle R_{H}}$ of ${\textstyle N_{t}}$=1,000 turns carrying a radio frequency current that generates a helicon^[58] wave that injects plasma fed from a source into a coil of radius ${\textstyle R_{c}}$ that carries a current of ${\displaystyle I_{c}}$, which generates a magnetic field. The excited injected plasma enhances the magnetic field and generates a miniature magnetosphere around the spacecraft, analogous to the heliopause where the Sun injected plasma encounters the interstellar medium, coronal mass ejections or the Earth's magnetotail. The injected plasma created an environment that analysis and simulations showed had a magnetic field with a falloff rate of ${\displaystyle 1/r}$ as compared with the classical model of a ${\textstyle 1/r^{3}}$ falloff rate, making the much smaller coil significantly more effective based upon analysis^[59] and simulation.^[8] The pressure of the inflated plasma along with the stronger magnetic field pressure at a larger distance due to the lower falloff rate would stretch the magnetic field and more efficiently inflate a magnetospheric bubble around the spacecraft.

Parameters for the coil and solenoid were ${\displaystyle R_{H}}$=2.5 cm and for the coil ${\textstyle R_{c}}$= 0.1 m, 6 orders of magnitude less than the magsail coil with correspondingly much lower mass. An estimate for the weight of the coil was 10 kg and 40 kg for the plasma injection source and other infrastructure. Reported results from Figure 2 were ${\displaystyle B_{0}\approx }$ ×10⁻³ T at ${\displaystyle R_{mp}\approx }$ 10 km and from Figure 3 an extrapolated result with a plasma injection jet force ${\textstyle F_{jet}\approx }$10⁻³ N resulting in a thrust force of ${\textstyle F_{M2P2}\approx }$ 1 N. The magnetic-only sail force from equation MHD.3 is ${\displaystyle F_{w}}$=3x10⁻¹¹ N and thus M2P2 reported a thrust gain of 4x10¹⁰ as compared with a magnetic field only design. Since M2P2 injects ionized gas at a ${\displaystyle m_{in}}$ mass flow rate (kg/s) it is viewed as a propellant and therefore has a specific impulse (s) ${\displaystyle I_{sp}=F_{M2P2}/m_{in}/g_{0}}$ where ${\displaystyle g_{0}}$ is the acceleration of Earth's gravity. Winglee stated ${\displaystyle m_{in}}$=0.5 kg/day and therefore ${\displaystyle I_{sp}}$=17,621. The equivalent exhaust velocity ${\displaystyle v_{e}=g_{0}\,I_{sp}}$ is 173 km/s for 1 N of thrust force. Winglee assumed total propellant mass of 30 kg and therefore propellant would run out in 60 days.^[8]