Editing Magnetic sail (section)

==== MHD model ====
Analysis of magsail performance was done using a simulation and a fluid (i.e., MHD) model with similar results observed for one case.<ref name=":13" /> The [[Magnetic moment#Amperian loop model|magnetic moment of a current loop (A m<sup>2</sup>)]] is <math>\mathbf m= I_c \pi R_c^2</math>  for a current of <math display="inline">I_c</math> A and a loop of radius <math display="inline">R_c</math> m.  Close to the loop, the magnetic field at a distance <math>z</math> along the center-line axis perpendicular to the loop is derived from the [[Biot-Savart law]] as follows.<ref>{{Cite web |last=Zhan |first=Marcus |date=2003 |title=Electromagnetic Field Theory: A Problem Solving Approach |url=https://ocw.mit.edu/courses/res-6-002-electromagnetic-field-theory-a-problem-solving-approach-spring-2008/pages/textbook-contents/ |access-date=July 3, 2022 |website=cow.mit.edu}}</ref>{{Rp|location=sec 5-2, Eq (25)}}

{{NumBlk2|:|<math display="block">B_{cl} (z)=\frac {\mu_0 I_c R_c^2}{2(z^2 +R_c^2)^{3/2} }</math>|MS.1}}

At a distance far from the loop center the magnetic field is approximately that produced by a [[Magnetic dipole#External magnetic field produced by a dipole moment|magnetic dipole]]. Te pressure at the magnetospheric boundary is doubled due to compression of the magnetic field and stated by the following equation at a point along the center-line axis or the target magnetopause standoff distance <math>L_Z</math>.<ref name=":13" />{{Rp|location=Eq (5)}}{{NumBlk2|:|<math>p_{mb}=\frac {B_{cl}(0)^2}{2 \mu_0} \Biggl(\frac {R_c}{L_Z}\Biggr)^6</math>|MS.2}}
Equating this to the dynamic pressure for a plasma environment <math display="inline">p_{mb}=\rho \, u_{pe}^2 /2</math>, inserting <math display="inline">B_{cl}(0)</math> from equation {{EquationNote|MS.1}} and solving for <math>L_Z</math> yields<ref name=":13" />{{Rp|location=Eq (6)}}
{{Numbered block 2|:|<math>L_Z=1.26 \, C_Z, where \, \, C_Z = R_c \Biggl( \frac {B_{cl}(0)}{u_{pe} \sqrt{\rho \mu_0} } \Biggr)^{1/3}</math>|MS.3}}

Andrews and Zubrin derived the drag (thrust) force of the sail <math>F_D</math><ref name=":13" />{{Rp|location=Eq (8)}} that determined the characteristic length <math>L_Z</math> for a tilt angle, but according to Freeland<ref name=":14" />{{Rp|location=Sec 6.5}} an error was made in numerical integration in choosing the ellipse downstream from the magnetopause instead of the ellipse upstream that made those results optimistic by a factor of approximately 3.1, which should be used to correct any drag(thrust) force results using<ref name=":13" />{{Rp|location=Eq 8}} Instead, this article uses the approximation<ref name=":14" />{{Rp|location=Eq (108)}}  for a spherical bubble that corrects this error and is close to the analytical formula for the axial configuration as the force for the Magsail as follows{{NumBlk2|:|<math>F_{MS} = 1.214 \, \,  \frac{\rho u_{pe}^2}{2} \pi \, L_Z^2</math>|MS.4}}
In 2004 Toivanen and Janhunen did further analysis on the Magsail that they called a Plasma Free MagnetoPause (PFMP) that produced similar results to that of Andrews and Zubrin.<ref name=":8" />