Achieving relativistic speeds by lasers

Question

Basic data about our spacecraft:

Basically were using a laser as an engine.

1. Mass: 1000 Tons
2. Electrical Power for the laser: 10 GW
3. Efficiency of the laser(s) : 50%

So first I calculated the kinetic energy the laser would give to the ship:

E = P * t * Le | P = electrical power , t = time , Le = laser efficiency (0.5)

Then based of that i calculatecd velocity:

v = sqrt(E/(m/2))

If these calculations are correct , a 1000 Ton ship , will reach 6000 m/s after 1h of acceleration using that engine.

The 10GW could be generated by a fusion reactor e.g.

The reasons why i wouldnt use the fusion reactor directly as an engine is 1. the nuclear waste it produces , so we couldnt use it around earth. 2. the exhaust velocity isnt c , the max top speed will be lower than 99.9c

My question is , if 1. my calculations are correct , and 2nd if my assumption using the nuclear reactor as an engine directly will result in a lower top speed.

And is this engine a good option for reaching relativistic speeds?

Answer 1

For propulsion you have to think momentum rather than energy. In this case almost all of the optical energy output of the laser remains in the emitted photons. Whomever is unlucky enough to find themselves in that beam will tell you that, just before they vaporize!

So what's the momentum of a photon? Since photons don't have rest mass, a photon with energy $E$ has momentum $E/c$.

Thrust, which is momentum per second, would be energy per second (power) divided by $c$, or $P/c$ where $P$ is power in Watts.

If you have 5 GW of laser power then thrust would be 5E+09/3E+08 or 16.7 Newtons. Yep, it's very tiny.

You can then get your acceleration from $F=ma$ → $a=F/m$ which is tiny, but after months or years can add up. Final velocity would be $v=at$ after a time $t$ in seconds.

So in two years you get a little past 1 kilometer/second.

I'll leave that up to you if it's good or not for your application. If you don't mind waiting ten thousand generations, then this might be fine.