deletions | additions       diff --git a/section_Creation_of_a_Magnetosphere__.tex b/section_Creation_of_a_Magnetosphere__.tex  index 2b45bae..c6d3689 100644  --- a/section_Creation_of_a_Magnetosphere__.tex  +++ b/section_Creation_of_a_Magnetosphere__.tex 
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To shield Mars from the same amount of cosmic radiation as the Earth, we want Mars' magnetopause to be approximately 20 Mars radii away from its center: about the same distance from Earth's center to its magnetopause.  $R_{Magnetopause}=[2B_0^2/{\mu_0\rho_{sw}v_{sw}^2}]^{1/6}$ $R_{Magnetopause}=[frac{2B_0^2}{\mu_0\rho_{sw}v_{sw}^2}]^{1/6}$  where $B_0$ is the magnetic dipole moment of the planet and $\rho_{sw}$ and $v_{sw}$ are the density and velocity of solar wind at Mars' orbit respectively.  Because the convecting currents which generate the magnetosphere are complex and occur within the spherical outer core, we will simplify our calculations by approximating Mars' outer core as a single gigantic rotating cylinder with radius ${R_{outer core} - R_{inner core}}/2$ and height $2R_{outer core}$. With this approximation, we find Mars' magnetic dipole moment to be  $B_0 = {\rho_c\omega\mu_0}/{16\pi}({R_{outer \rho_c\omega\mu_0({R_{outer  core}-R_{inner core}}/2)^42R_{outer core}$ core}}/2]^42R_{outer core}/{16\pi}$  where $\rho_c$ is the charge density of the outer core and $\omega = {2\pi}/{2t_{con}}$ 2\pi/2t_"con"$  where $t_{con}$ $t_con$  is the convective timescale. $t_con$ is mathematically related directly to the adiabatic gradient, and using this we can find the necessary inner core temperature $T_{inner}$ $T_inner$  as a function of the the outer core temperature $T_{outer}$. $T_outer$.  Consolidating the necessary equations we find that  $t_{con} $$t_con  = [T_{outer}/g((dT/dr)_{ad}-\delta{T}/\delta{r})^{-1}]$ [T_outer/g((dT/dr)_ad-\delta{T}/\delta{r})^-1]$$  where the adiabatic gradient  $(dT/dr)_{ad} $(dt/dr)_ad  = {\gamma (\gamma  - 1}/\gamma T_{outer}/P_c 1)/\gamma T_outer/P_c  dP/dr$ and $dP/dr = -4/3\piG\rho^2R_inner$, $g=4/3\piG\rhoR_inner$, and the planet's central pressure $P_c = 2/3\piG\rho^2R_p^4$.