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Whenever terraforming Mars comes up, the subject of its magnetosphere inevitably arises. When that happens, someone is bound to suggest using nukes to restart the core a la "The Core".
Please show proof.
According to Wikipedia, Mars has a differentiated iron-nickel core. It's radius is approximately 1800 km. This gives it a volume of $2.4\times10^{19}$ m$^2$. Its temperature is not well known, but some estimates are up to 1500 K. It is at least partially molten, so its temperature will depend on its sulfur content.
The Earth's liquid outer core has a density of around 10000 kg/m$^3$ and a temperature of at least 3000 K. Let's assume Mars' core has the same density of Earth's outer core, and needs to heat up by 1500 K to 'restart' it, whatever that means.
The specific heat of liquid iron is 820 J/kg/C; while Nickel is 730 J/kg/C. Let's estimate the specific heat of the core at 800 J/kg/C, overall. Multiply that by the volume, density and desired temperature differential (1500 K) to get a required energy of $2.9\times10^{29}$ J.
Using the best web page ever, we see that is roughly equal to the rotational energy of the Earth. It is also six orders of magnitude larger than the Chicxulub meteor (that killed the dinosaurs) and 12 orders of magnitude larger than the biggest nuclear weapon. So, you would need around a trillion of the largest nuclear weapons ever to add that much heat to the core.
Just for an additional frame of reference: If we put the Moon (like, Luna, our moon) on a Hohmann transfer orbit to intercept Mars, it would hit Mars at around 3 km/s. The kinetic energy of this impact is $3\times10^{29}$ J; this is the amount of energy needed to heat up the core as desired. Of course, this is ignoring all the heat energy it takes to heat up the mantle and the crust, the energy lost blasting bits into deep space, the energy lost through momentum transfer to Mars' new orbit....
It doesn't matter the current state of Mars' core, or if it is even possible to 'restart' it to the point that it will generate a protective magnetic field. It is not feasible add that much heat with nuclear weapons, nor is it feasible by hitting Mars with the Moon. It's not going to happen.
The problem with this idea is that nuclear bombs can't just start a core spinning inside the planet again, regardless of what Hollywood says; the science just isn't correct.
In the movie the Earth's core (made mostly of iron) had stopped or slowed its spin within the centre of the planet. On Earth, it can spin because of the liquid outer core and the mantle.
The core of the Earth spinning is good for life on Earth because it's what generates the magnetic field around the Earth. That magnetic field protects us from most of the solar radiation emitted by the Sun, protecting us from what would ultimately be lethal radiation.
One theory about why the core of the Earth is spinning in the first place is the theory about how the moon was formed (massive collision of a Mars sized body with the Earth early in its formation) which if it was a glancing blow, could have potentially put a lot of angular momentum into the Earth and into the core in particular. In the movie though, it had slowed (and was stopping) and that meant that the magnetic field was degrading, meaning that they had to go in to basically kick-start it again, or get it spinning back at its original angular speed.
Loosely described, the idea is that detonating a bomb in the outer core gets the hot liquid spinning faster, which (hopefully) drags the outer core around with it via friction. But if that was the case, friction would have stopped the core spinning long ago. That means that the science can't be sound, and even if it was, the bomb's energy release is omni-directional, meaning that all you're likely to do is disrupt the flow of the outer core by generating a massive amount of turbulence.
Basically, it's a great sounding but REALLY bad idea.
But for the sake of argument...
Let's assume it WAS possible on the Earth. The actual inner core of the earth is actually smaller than the solid core in Mars, despite Mars being a smaller planet. That means that you've got a smaller amount of mantle between it and the crust. So, you've got a larger mass to get spinning from a standing (not slow like in the movie) start, you've got a thinner mantle and a crust which because of the collisions Mars has experienced, is even less even (in proportion to planet size) that the Earth has. Add to that, the Earth has a massive body of water to even out some of the pressure variations caused by internal nuclear explosions and to tame most of the volcanic eruptions that would occur as a result of the disruptions being caused inside the planet by the explosions.
From what I've read, Mars doesn't seem to have a liquid outer core per se (but happy to be proven wrong if others have more detailed articles available). That means that the mantle, which is also liquid because of heat, but probably more viscous and therefore more likely to cause friction, could be a reason why the core of Mars doesn't spin in the first place. To solve the problem more permanently, that would have to be solved and I'm not convinced it can be, even with future tech.
If you have the tech to tunnel in to the core of Mars, then may I suggest an alternate solution? You've still got some heat down there (around 1230 degrees Celsius) so perhaps your answer is a thermal generator (a power source that can generate electricity from heat) that could power a MASSIVE dynamo, converting that heat energy to a magnetic field artificially.
While neither option is what I would call 'plausible', it seems safer and more 'doable' to artificially generate a magnetic field from heat energy than to attempt to get a large core spinning from a standing start in a mantle that may not be able to support it with bombs that would probably disrupt the surface of the planet significantly as well.
Just saying.
The magnetosphere isn't likely to be restarted due to the effort and the time frame required. Even if you could melt the core with nukes, you can't make it spin which is required to generate a magnetic field. You need a moon to generate spin.
Scientists are looking at artificial magnetic satellites that will generate a large enough field and park it between Mars and the sun so Mars will be covered by its magnetic shadow.
See magnetic shield.
