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\section{Creation of a Magnetosphere}
In order to retain this atmosphere, we will need to create a magnetosphere to shield Mars from solar wind and harmful cosmic radiation. Assuming Mars has constant density and a solid inner core and liquid outer core proportionally identical to the Earth's, we can create a dynamo by detonating nuclear weapons in Mars' inner core, thus inducing convection in Mars outer core.
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To give Mars the same amount of protection from solar wind 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.
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$R_{Magnetopause}=(\frac{2B_0^2}{\mu_0\rho_{sw}v_{sw}^2})^{1/6}$
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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.
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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
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$B_0 = \frac{\rho_c\omega\mu_0}{16\pi}(\frac{R_{outer core}-R_{inner core}}{2})^4*2R_{outer core}$
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where $\rho_c$ is the charge density of the outer core and $\omega = 2\pi/2t_{con}$ where $t_{con}$ is the convective timescale.
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$t_{con}$ 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 $T_{outer}$.
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Consolidating the necessary equations we find that
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$t_{con} = \frac{T_{outer}}{g}[(\frac{dT}{dr})_{ad}-\frac{\delta{T}}{\delta{r}})^{-1}]$
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where the adiabatic gradient
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$(\frac{dT}{dr})_{ad} = \frac{\gamma-1}{\gamma} \frac{T_{outer}}{P_c} \frac{dP}{dr}$
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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 $P_c = \frac{2\pi}{3}G\rho^2R_p^4$.
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Plugging in all equations and solving for $T_{inner}$, we find that
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$T_{inner} > 1.035T_{outer}$
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Plugging in $T_{outer} ~= 2400K$, we see that we will need a minimum temperature of $T_{inner} ~= 2500K$ in order to induce convection in the outer core and create a magnetic field.
