Editing Magnetic sail (section)

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

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

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

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

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

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

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

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

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

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

Using <math>f_o</math>=1 creates very optimistic performance numbers, but since Slough changed this to <math>f_o</math>=2 in 2011<ref name=":22" /> and 2012,<ref name=":7" />  the case of <math>f_o=1</math> is not compared in this article. An example for <math>f_o</math>=2 using solar wind parameters <math>\rho</math>=8x10<sup>−21</sup> kg/m<sup>3</sup>, <math>u</math>=500&nbsp;km/s then <math>r_g</math>=72&nbsp;km and <math>B_{mp}</math>=4x10<sup>−8</sup> T with <math>R_{mp}</math>=10<sup>5</sup> m results in <math>r_g < L</math> where MHD applicability occurs.  With a coil radius of <math>R_c</math>=1,000 m yields <math>B_0</math>=4x10<sup>−4</sup> T from equation {{EquationNote|MFM.2}}. The required RMF power from equation {{EquationNote|PM.3}} is 13&nbsp;kW with a required AC coil current <math>I_c</math>=10 A from equation {{EquationNote|PM.3}} resulting in an induced current of <math>I_{ic}</math>=2 kA from equation {{EquationNote|PM.7}} . With <math>C_d</math>=5 the plasma magnet force from equation {{EquationNote|PM.3}}  is 197 N. The magnetic force only for the above parameters is 2.8 N from equation {{EquationNote|MFM.5}} and therefore the plasma magnet thrust gain is 71. The [[#Performance comparison|performance comparison]] section gives and optimistic estimate using constant acceleration for  <math>f_o</math>=2 results in a transit time of ~100 days.<!-- DM: Confirm that parameter choices in the para match PM in comparison table. -->