Does the sun provide enough energy to accelerate a large ship to a decent proportion of the speed of light?

Question

This came up when running some numbers on the hard science possibilities of interstellar colonization and I mostly want to know whether I made some major errors and am off by a few orders of magnitude somewhere.

First, assuming the speed of light is an absolute bound, any possible space ship needs to be completely autark, that is have everything it and its passengers will ever need on board. So it needs to be pretty big. I will just assume a cube of 10km edge length. Putting a density of 1 (high density building material filled with air) this gives a mass of $10^{12}$ tons.

Second, if you want to get to any other stars within human life spans you need a speed that is a decent proportion of the speed of light, say $0.5c$.

To compute the fuel needs to accelerate $10^{12}$ tons to $0.5c$ (and then decelerate to zero again) we use the rocket equation, for example here. An exhaust velocity of $1c$ and a target speed of $0.5c$ gives us that we need approximately as much fuel as the mass we want to accelerate (the calculator gives $40\%$ fuel and $60\%$ ship weight).

To achieve an exhaust velocity of $1c$ we will use an antimatter drive. The antimatter will be made from the energy from our sun. The total energy output of the sun corresponds to converting $4*10^6$ tons of matter to energy per second. So if one is able to capture the entire energy output of the sun, one would need about $10^{12}/(4*10^6)=2.5*10^5$ seconds or around 7 hours.

Meaning in summary accelerating a single ship of this size to around half the speed of light would already take the complete energy output of our sun for a several hours assuming $100\%$ perfect efficiency in every step. So we are talking a full Dyson sphere, not just some large solar sails. Does that make sense of did I miss something major somewhere?

Edit (in response to comments): I was indeed thinking of a fully self sustaining colony ship and that does require millions of people on there. The entire computation is also linear in the mass, so if you are interested in the numbers for a smaller ship, you can simple divide the needed energy by the same factor that you reduce the mass. Similary $100\%$ efficiency is easiest to look at because one can account for lower efficiency by simply multiplying in whatever efficiency one wants to use.

Answer 1

You've got a few minor errors in there... it is easy to do, and a good reason for doing all your working in a program or spreadsheet or a mathmatical modelling tool or whatever. The obvious mistake is 250000 seconds is 70 hours, not 7, and you're applying your mass-creation rate to the total mass of the ship.

Here are my workings and results:

For a delta-V of 1c and an exhaust velocity of 1c (and ignoring relativistic effects, which is slightly naughty but not too naughty at only .5c max velocity) you end up with a mass ratio of e (~2.7), this is because the rocket equation gives us $R = e^\frac{\Delta_v}{V_e}$. If your total launch mass is 10¹² tonnes (I'll use the metric ones here), that means you'll have a dry mass of 10¹²/$e$, or ~3.7x10¹¹ tonnes, NS ~6.32x10¹¹ tonnes of fuel. That's more like 63.2% fuel, 36.8% ship.

(I note that if you had a delta-V of 0.5c, so only enough to get up to cruising speed and not to slow back down again, you do end up with a mass ratio of $\sqrt{e}$, which would be reflected as a launch that was 60% ship and 40% fuel, which were the figures you originally came up with.)

Your fuel will only be 50% antimatter. This makes the next bit easier, because when you're conjuring up matter ex nihilo, you end up with equal amounts of matter and antimatter because of the conservation of baryon number.

Assuming a perfect conversion of energy to mass, e=mc² shows you need 6.32x10¹¹ x c² or ~5.68x10³¹J of energy. The sun's luminosity is 3.828x10²⁶W. If you harvest 100% of that, it'll take you ~148412 seconds (or a little over 41 hours) to create the required mass of matter and antimatter.

If you meant short ton instead of metric tonne, your mass requirements are dropped slightly, and it will take more like ~134637 seconds to generate the required matter and antimatter, or about 37.4 hours.

There are better ways to do what you want to do (for a start, using magnetic braking at your destination instead of a bajillion tonnes of antimatter) but that's a bit outside of the scope of this question. You might ask a different question about how to reduce fuel demands without compromising journey time.