deletions | additions       diff --git a/section_Creation_of_a_Magnetosphere__.tex b/section_Creation_of_a_Magnetosphere__.tex  index d72cf00..f2d9761 100644  --- a/section_Creation_of_a_Magnetosphere__.tex  +++ b/section_Creation_of_a_Magnetosphere__.tex 
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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\\  $B_0 = \frac{\rho_c\omega\mu_0}{16\pi}(\frac{R_{outer core}-R_{inner core}}{2})^4*2R_{outer core}$\\  where $\rho_c$ is the charge density of the outer core and $\omega = 2\pi/2t_{con}$ where $t_{con}$ 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 $T_{inner}$ as a function of the the outer core temperature $T_{outer}$. $T_{outer}$.\\  \\  Consolidating the necessary equations we find that that\\  \\  $t_{con} = \frac{T_{outer}}{g}[(\frac{dT}{dr})_{ad}-\frac{\delta{T}}{\delta{r}})^{-1}]$ \frac{T_{outer}}{g}[(\frac{dT}{dr})_{ad}-\frac{\delta{T}}{\delta{r}})^{-1}]$\\  \\  where the adiabatic gradient gradient\\  \\  $(\frac{dT}{dr})_{ad} = \frac{\gamma-1}{\gamma} \frac{T_{outer}}{P_c} \frac{dP}{dr}$ \frac{dP}{dr}$\\  \\  and $\frac{dP}{dr} = -\frac{4\pi}{3}G\rho^2R_{inner}$, $g = \frac{4\pi}{3}G\rho^2R_{inner}$, and the planet's central pressure $P_c = \frac{2\pi}{3}G\rho^2R_p^4$. \frac{2\pi}{3}G\rho^2R_p^4$.\\  \\  Plugging in all equations and solving for $T_{inner}$, we find that that\\  \\  $T_{inner} > 1.035T_{outer}$  \\