Editing Magnetic sail (section)

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

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

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

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

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

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

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

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

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