Editing Magnetic sail (section)

=== Magnetic field model ===
In a design, either the magnetic field source strength or the magnetopause radius <math>R_{mp}\approx L</math> the characteristic length must be chosen. A good approximation from Cattell<ref name="05ja026_full" /> and Toivanen<ref name=":8" /> for a magnetic field falloff rate <math>f_o,  1\leq f_o \leq3 </math> for a distance <math display="inline">R_0 \leq r \leq R_{mp} </math> from the field source to magnetopause starts with the equation:
{{NumBlk2|:|<math>B(r)\approx B(R_0)\biggl(\frac {R_0}{r}\biggr)^{f_o}</math>|MFM.1}}

where <math>B(R_0) </math> is the magnetic field at a radius <math>R_0</math> near the field source that falls off near the source as <math>1/r^3</math> as follows:

{{NumBlk2|:|<math>B(R_0) \approx C_{0} \frac {\mu_0\, \mathbf m }{4 \, \pi \, R_0^3}</math>|MFM.2}}

where <math>C_0</math> is a constant multiplying the [[magnetic moment|magnetic moment (A m<sup>2</sup>)]]   <math>\mathbf m</math> to make <math>B(r)</math> match a target value at <math display="inline">r>>R_0</math>. When far from the field source, a magnetic dipole is a good approximation and choosing the above value of <math>B(R_0) </math> with <math>C_0</math> =2 near the field source was used by Andrews and Zubrin.<ref name=":13" />

The [[Magnetic moment#Amperian loop model|Amperian loop model]] for the magnetic moment is  <math>\mathbf m = I_c S</math>, where <math>I_c</math>  is the current in [[Ampere|amperes (A)]] and <math>S= \pi \, R_c^2</math> is the surface area for a coil (loop) of radius <math>R_c</math>. Assuming that <math>R_0 \approx R_c</math> and substituting the expression for the magnetic moment <math>\mathbf m</math> into equation {{EquationNote|MFM.2}} yields the following:{{NumBlk2|:|<math>I_c=\frac {4}{C_0 \mu_0} B(R_c) \, R_c</math>|MFM.3}}

When the magnetic field flux density <math>B(R_0) </math> is specified, substituting <math>B_{mp} </math> from the pressure balance analysis from equation {{EquationNote|MHD.2}} into the above and solving for <math display="inline">R_{mp} </math> yields the following:{{NumBlk2|:|<math>L \approx R_{mp} \approx R_0  \biggl(\frac {B(R_0)}{B_{mp} } \biggr)^{1/f_o} </math>|MFM.4}}

This is the expression for <math>L</math> when <math>f_o=3 </math> with <math>C_{SO}=1/2 </math><ref name=":3" />{{Rp|location=Eq (4)}} and <math>C_{SO}=2 </math><ref name=":8" />{{Rp|location=Eq (4)}} and is the same form as the [[Magnetopause#Magnetopause distance of the Earth|magnetopause distance of the Earth]]. Equation {{EquationNote|MFM.4}} shows directly how a decreased falloff rate <math>f_o </math> dramatically increases the effective sail area <math>S=\pi R_{mp}^2 </math> for a given field source magnetic moment <math>\mathbf m</math> and <math>B_{mp}</math> determined from the pressure balance equation {{EquationNote|MHD.1}}. Substituting this into equation {{EquationNote|MHD.3}} yields the plasma wind force as a function of falloff rate <math>f_o </math>, plasma density <math>\rho</math>, coil radius <math>R_c \approx R_0</math>, coil current <math>I_c</math> and plasma wind velocity <math>u</math> as follows:{{NumBlk2|:|<math>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}{4 R_0 \, u \sqrt{\rho \,C_{SO} } } \biggr)^{2/f_o}
</math>|MFM.5}}

using equation {{EquationNote|MFM.3}} for <math>B(R_0) </math> and equation {{EquationNote|MHD.2}} for <math>B_{mp}</math>. This is the same expression as equation (10b) when <math>f_o=3 </math> and <math>C_{SO}=1/2 </math><ref name=":0" /> and<ref name=":14" />{{Rp|location=Eq (108)}} and the right hand side from equation (20) specifically applied to M2P2<ref name=":8" /> with other numerical coefficients grouped into the <math>C_d</math> term.  Note that force increases as falloff rate decreases. For the solar wind case, substituting {{EquationNote|MHD.2}} into {{EquationNote|MFM.5}} and using the function for the [[#Acceleration/deceleration in a stellar plasma wind|solar wind plasma mass density]] <math>\rho_{sw}(a_\odot)=\rho_{sw}(1)/a_\odot ^2</math>,<ref name=":32" />{{Rp|location=Fig 5}} with <math>a_\odot

</math> the distance from the sun in Astronomical units (AU) results in the following expression: {{NumBlk2|:|<math>F_{sw}(f_o, a_\odot)=C_{f_o} \, ( \rho_{sw}(a_\odot) \, u^2))^ {1-1/f_o} \,

R_0^2 \, S  
</math>|MFM.6}}where <math>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

</math>, the effective sail blocking area.

This equation explicitly shows the relationship upon solar wind plasma mass density <math>\rho_{sw}(a_\odot)

</math> as a function of distance from the Sun <math>a_\odot

</math>.  For the case <math>f_o</math>=1 the expansion of the magnetopause radius <math>R_{mp}

</math> exactly matches the decreasing value of <math>\rho_{sw}(a_\odot)

</math>  exactly as the distance from the Sun <math>a_\odot

</math> increases, resulting in constant force and hence constant acceleration inside the heliosphere.<ref name="Slough2006" /> Note that <math>C_{f_o}</math> includes the term <math>B(R_0)^{2/f_o}</math>, which means that as <math>f_o</math> increases that the magnetic field near the field source <math>B(R_0)</math> must increase to maintain the same force as compared with a smaller value of <math>f_o</math>. The example in the [[#Overview|overview]] section set <math>C_{f_o}</math>=1, <math>u</math>=1, <math>R_0</math>=1, and <math>S</math>=1 so that the force at <math>a_\odot

</math>=1 was equal to 1 for all values of <math>f_o</math> at 1 AU.