Editing Magnetic sail (section)

== Performance comparison ==
The table below compares [[#Performance measures|performance measures]] for the magnetic sail designs with the following parameters for the solar wind (sw) at 1 AU: velocity <math>u_{sw}</math>= 500&nbsp;km/s, number density <math>n_i</math>= 5x10<sup>6</sup> m<sup>−3</sup>, ion mass <math>m_i</math> = 1.67x10<sup>−27</sup> kg a [[proton mass]], resulting in mass density <math>\rho_{sw}= m_i n_i</math> = 8.4x10<sup>−21</sup> kg/m<sup>3</sup>,  and coefficient of drag <math>C_d</math>=5 where applicable. Equation {{EquationNote|MHD.2}} gives the magnetic field at magnetopause as <math>B_{mp}</math>≈ 36 nT, equation {{EquationNote|MHD.5}} gives the ion gyroradius <math>r_g </math>≈ 72&nbsp;km for <math>C_{Li}</math>=2. Table entries in boldface are from a cited source as described in the following:

Equation {{EquationNote|MS.4}} determines force for the [[#Magsail (MS)|Magsail (MS)]] divided by the Freeland correction factor 3.1,<ref name=":14" /> equation {{EquationNote|PM.5}} defines the force for the [[#Plasma magnet (PM)|plasma magnet (PM)]] with the assumed magnetic field falloff rate <math>f_o</math>=2. The force for the magnetic sail alone <math>F_{mag}</math> is from equation {{EquationNote|MFM.5}}. Thrust gain <math>G_T</math> for the [[#Magnetoplasma sail (MPS)|magneto plasma sail (MPS)]] is the simulation and/or experimentally determined value with force defined equation {{EquationNote|MPS.2}} to account for thrust loss due to operation in a kinematic region. The last column headed MPS+MPD adds a [[Magnetoplasmadynamic thruster|magnetoplasma dynamic thruster]] (MPD) that has a higher thrust gain as determined by experiment and simulation. Further details are in the section for the specific design. For designs other than MPS and MPS+MPD, the thrust gain <math>G_T</math> is the achieved force from the first row divided by the force of a magnetic sail alone in the second row. The magnetopause distance <math display="inline">R_{mp} \approx L</math> and the coil radius <math display="inline">R_c</math> are design parameters. Equation {{EquationNote|MFM.1}} with <math display="inline">r=R_{mp} </math> defines the magnetic field near the coil(s) as <math display="inline">B_0=B(R_0)=B_{mp}(R_{mp}/R_0)^{1/f_o}</math>.

The superconducting coil designs used a critical current density <math>J_c</math>=2x10<sup>6</sup> A/m to account for warmer temperatures in the solar system. The plasma magnet uses AC power for the rotating magnetic field, P<sub>RMF</sub> as specified in Equation {{EquationNote|PM.3}} and cannot use a superconducting coil and assumed an aluminum coil with material density <math>\delta_c</math> = 2,700&nbsp;kg/m<sup>3</sup> and coil wire radius <math>r_c</math>=5&nbsp;mm.  All other designs assumed a superconducting coil with material density <math>\delta_c</math> =6,500&nbsp;kg/m<sup>3</sup>,  coil wire radius <math>r_c</math>=5&nbsp;mm,  and critical current <math>\max I_c \approx </math>1.6 x10<sup>6</sup> A, above which the coil becomes a normal conductor. The magnetopause distance <math>R_{mp} \approx L</math> and coil radius <math>R_c</math> for superconducting-coil based designs were adjusted to meet this critical current constraint. The values for the plasma magnet used a value of <math>R_c</math> for <math>f_o</math>=2 selected to minimize time to velocity and distance. The MPS values for <math>R_{mp} \approx L</math> and <math>R_c</math> were chosen to match the thrust gain from simulation and scaled experimental results and meet the superconducting-coil critical current constraint.

Equation {{EquationNote|CMC.2}} gives the physical coil mass <math>M_c(phy)</math> assuming a coil wire radius <math>r_c</math>=5&nbsp;mm. Equation {{EquationNote|PM.7}} gives the plasma magnet alternating current <math>I_c </math>.  Equation {{EquationNote|MFM.3}} gives the direct current <math>I_c </math> with <math>C_0</math>=2 for all other designs. The plasma magnet RMF uses the input alternating current <math>I_c</math> (kA) to rotate electrons in captured plasma to create an induced direct current disc carrying <math>I_{ic}</math> kA as defined in equation {{EquationNote|PM.8}}.

Superconducting coils do not require continuous power (except possibly for cooling); however, the plasma magnet design does, as specified in equation {{EquationNote|PM.3}}. An estimate for the plasma magnet power supply mass assumes ~3&nbsp;kg/W for [[nuclear power in space]]. Other mass was assumed to be 10 tonnes for MS and 1 tonne for PM and MPS. Acceleration <math>a</math> is the thrust force <math>F</math> from the first row divided by the total mass (coil plus other). An optimistic approximation is constant acceleration <math>a</math>, for which the time to reach a target velocity V of 10% of the solar wind velocity is <math>T_V \approx V/a</math> and time to cover a specified distance <math>D</math> ≈ 7.8x10<sup>8</sup> km (approximate distance from Earth to Jupiter) is <math display="inline">T_D \approx \sqrt{ 2D/a}</math> . For comparison purposes the time for a [[Delta-v budget#Interplanetary|Hohmann transfer]] 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.<ref name=":11" /> Another comparison is the [[New Horizons]] interplanetary space probe with a 30&nbsp;kg payload that flew by Jupiter after 405 days on its way to Pluto.
{| class="wikitable"
|+
!Parameter
!Description
!Magsail
!PM, f<sub>o</sub>=2
!MPS
!MPS+MPD
!Units
|-
|<math>F</math>
|Thrust force
|644
|197
|700
|611
|N
|-
|<math>F_{mag}</math>
|Magnetic Force
|644
|2.8
|175
|51
|N
|-
|<math>f_o</math>
|Falloff rate
|'''3'''
|'''2'''
|'''2'''
|'''2'''
|
|-
|<math>G_T</math>
|Thrust gain
|1
|71
|'''4'''
|'''12'''
|
|-
|<math>L, R_{mp}</math>
|Magneto-pause
|'''520'''
|100
|100
|60
|km
|-
|<math>R_c</math>
|Coil radius
|'''100,000'''
|1,000
|6,300
|2,900
|m
|-
|<math>B(R_c)</math>
|Coil field
|2.5x10<sup>−6</sup>
|4x10<sup>−4</sup>
|1.6x10<sup>−4</sup>
|3.5x10<sup>−4</sup>
|T
|-
|<math>I_c</math>
|Coil current
|405
|0.01
|1,593
|1,624
|kA
|-
|<math>I_{ic}</math>
|Induced current
|—
|2
|—
|—
|kA
|-
|<math>P_O</math>
|Required power
|—
|13
|—
|—
|kW
|-
|<math>M_c</math>
|Coil mass
|'''474'''
|4
|30
|14
|tonnes
|-
|<math>M_O</math>
|Other mass
|10
|5.3
|1
|1
|tonnes
|-
|<math>a</math>
|Acceleration
|0.0013
|0.021
|0.023
|0.042
|m/s<sup>2</sup>
|-
|<math>T_V</math>
|Time to velocity
|435
|27
|26
|14
|days
|-
|<math>T_D</math>
|Time to distance
|396
|99
|96
|71
|days
|}
The best time to velocity <math>T_V</math> and distance <math>T_D</math> performance occurs for the PM and MPS designs due primarily to much reduced coil and other mass. As described in the [[#Mini-magnetospheric plasma propulsion (M2P2)|M2P2 section]], several criticisms asserted that the falloff rate <math>f_o</math>=1 was questionable and hence it was not included in this table. Simulations and experiments as described in the [[#Magnetoplasma sail (MPS)|MPS section]]  showed that <math>f_o</math>=2 is valid with injection of plasma to inflate the magnetic field in a manner similar to M2P2.  As described in the [[#Plasma magnet (PM)|PM section]], plasma is not injected but instead captured to achieve a falloff rate of <math>f_o</math>=2,<ref name=":7" /> with calculations assuming <math>f_o</math>=1 being very optimistic. The classic [[#Magsail (MS)|Magsail (MS)]] design generates the most thrust force and has considerable mass but still has relatively good time performance. Parameters for the other designs were chosen to yield comparable time performance subject to the constraints previously described. As 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.