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I think that other than in a star, nobody has achieved any fusion process with a net positive energy output.

But reading about the Bussard ramjet a few minutes ago, it seems that this might be theoretically possible given adequate speed.

Has anyone calculated the minimum velocity necessary for a Bussard ramjet to be able to achieve hydrogen fusion with a net positive energy output?

Some assumptions would obviously be necessary (average hydrogen density et. al.), but this seems like it would be a fun calculation. Though it would probably require at least days for me to try.

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The key part to answer this question can be found at dangermouse.net. Specifically, this formula must be satisfied:

$V_t > \frac{dm}{dt} / ( π\cdot r2 \cdot\ ρ)$

Where $V_t$= Velocity, $\frac{dm}{dt}$ is the minimum fusion rate of hydrogen, $ π\cdot r2 $ is the area of the funnel, and $ρ$ is the density of hydrogen in the interstellar medium. Right now, all of these are known except for one, namely what is the minimum rate to fuse hydrogen. This will very likely depend on the engine. I'm going to say that we need to collect 1 gram every second, just as a demonstration, feel free to insert the real requirement when you have it. Let's also assume a 1 km funnel, as that seems to be a popular size for a rotational sphere. Let's assume about 20 atoms/cm^3, per Wikipedia. That gives us a required speed of:

$(1)/(10^6*pi*20e6/6.0221413e+23)=9584535558.93 m/s=31c$

which is impossible, being 31 times faster than the speed of light.

The bottom line is, this would require a huge funnel, and the ability to fuse hydrogen at trace quantities, or operating in a dense cloud of hydrogen, in order to make it work. For the mean time, it seems extremely difficult.

Given the same set of numbers, let's see what a more realistic 20 AU/year, or about 100 km/s. This chosen because it's the fastest conventional rocket I've heard of. That would require a funnel area of:

$(1)/(100000 \cdot 20e6/6.0221413e+23)=301107064500 m^2$ That leaves a radius of 175 km, which is clearly large, but not impossible!

Let's try one more time, with the solar escape velocity from the sun's surface, and the particle density near the sun. That density is about $10^9 particles/m^2$, solar escape velocity is 617 km/s. Thus, the required size is:

$(1)/(617000 \cdot 1e9/6.0221413e+23)=976035867.099m^2$ That leaves a radius of 10 km. Thus, the best thing to do with a ramjet is to somehow have it pass right by the sun. Activating the ram jet there would allow one to gain some speed. I'm not going to go through the math to determine the appropriate acceleration to make it work at that speed, but it could work, given appropriate amounts of mass.

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