# Mass required to accelerate to relativistic speed

What would the rest mass of a rocket need to be if it carries a payload of 1kg and is accelerated to 1/2 c and then decelerated to a stop. Assume: 1 - energy required is obtained via fusion of hydrogen fuel carried on board

Second question: Alternatively, what would the rest mass need to be if energy required is obtained via matter/anti-matter annihilation?

• Although it doesn't directly answer your questions, I think you would enjoy the relativistic rocket page. – PM 2Ring Jun 9 at 4:12
• I calculate that you need 2kg of matter+antimatter fuel per kg of the rocket's final mass, assuming a perfect engine. Don't forget to include the "dry" mass of the rocket in the payload. See physics.stackexchange.com/a/345492/123208 for details. – PM 2Ring Jun 9 at 4:50
• Have you checked out worldbuilding.stackexchange.com? – lsusr Jun 9 at 8:30

# Hydrogen Engine

The answer to this question depends on what kind of hydrogen you use.

"Hydrogen" usually refers to hydrogen-1. If you have the technology to fuse hydrogen-1 then you don't need to carry it with you because you can collect it with a ramscoop. There's no theoretical lower limit to the mass of a ramscoop. But this question specifically asks about hydrogen fuel "carrried on board".

If you're bringing the hydrogen with you then you shouldn't be bringing hydrogen-1. Hydrogen-1 is the hardest kind of hydrogen to fuse and has a lower ratio of energy released to rest mass than deuterium-tritium fusion. It'd be better to bring hydrogen-2 (deuterium) and/or hydrogen-3 (tritium). The answer to your question depends on whether you're fusing hydrogen-1, hydrogen-2, hydrogen-3 or some combination thereof. They all release different quantities of energies when fused.

Once you know how efficient your hydrogen fusion is the next question is what you use as exhaust. If you have the technology you could build a photonic rocket. Otherwise your answer will depend on the rocket's exhaust speed. Faster exhaust = faster rocket. Your Lorentz factor at $$\frac12 c$$ is only 1.15 so you can get a reasonable estimate from this classical model.

But really, you should be using a ramscoop or a laser-assisted solar sail. Not hydrogen-1 carried on board.

Edit: @dmckee notes that a naive ramscoop will only bring you up to 14% $$c$$ due to drag. The remaining 36% must be obtained from other means.

# Matter-Antimatter Engine

Any civilization advanced enough to build a matter/anti-matter engine also has the technology to build a photonic rocket. A photonic rocket is the most efficient type of rocket. Instead of emitting water vapor like our chemical rockets it emits photons. This is important because exhaust velocity is the most important factor in determining the efficienty of rockets. Faster exhaust velocity equals faster rockets. Nothing goes faster than light.

Conceptually, the easiest way to think about this question is as a two-stage photon rocket. One photon rocket with a $$\Delta v=\frac12$$ contains a second photon rocket rocket with $$\Delta v=\frac12$$.

The speed of each photon rocket is as follows.

$$\frac12 c=\Delta v=c\frac{\left(\frac{m_i}{m_f}\right)^2-1}{\left(\frac{m_i}{m_f}\right)^2+1}$$

Solving for $$\frac{m_i}{m_f}$$ we get $$\frac{m_i}{m_f}=\sqrt 3\approx 1.732$$.

Since this is a two-stage rocket we apply the equation twice, recursively. Let's call $$m_2$$ the initial rocket (both stages), $$m_1$$ the mass of the entire second stage and $$m_0$$ the payload.

$$\frac{m_2}{m_1}=\frac{m_1}{m_0}=\sqrt 3$$ $$\frac{m_2}{m_0}=3$$

Therefore you must expend 2kg of matter+antimatter fuel to accelerate 1 kg to 1/2 c and then decelerate it back to rest. Total rest mass = 3 kg.

• Keep in mind that a naive hydrogen ram-scoop can only achieve about 14% c. Here "naive" means that you have to bring the scooped material to relative rest in order to manipulate it into fusing, which implies a large drag. Assume that this requirement may be partially violated by clever technology and you get numbers like the 0.2 c that Vinge used for his Queg Ho. – dmckee Jun 9 at 17:43
• Could we halve the fuel load in the antimatter case by only bringing the antimatter? We could collect the matter to annihilate on the way. Though this introduces some of limitations of the other case instead. – badjohn Jun 10 at 7:26
• @badjohn Yes you can. – lsusr Jun 10 at 19:36