[Spaceships] Travel times using fusion rockets

[Anthony]

Originally Posted by Michael Thayne

It isn't too hard to do the calculations for slowly boosting out of LEO using a fusion rocket.

It's hard to do without a computer.

[Michael Thayne]

Originally Posted by Anthony

It's hard to do without a computer.

That's not much of a barrier if you're the kind of person who hangs out on internet forums. Even if you don't have a fancy pants MacBook, you can do crude simulations in Google Sheets. I understand many people don't like laptops at the gaming table, but I'd want to pre-compute this stuff anyway.

[scc]

Originally Posted by Michael Thayne

It isn't too hard to do the calculations for slowly boosting out of LEO using a fusion rocket. The results I get are that at 0.005G, it takes a little over three days, and you burn 8.3 mps delta-V, but are only traveling at 5.8 mps when you cross the edge of Earth's SOI. Here's the Ruby code for that simulation (using one-second steps):

fusion_rocket_acceleration_mms = 0.005 * 9.81
earth_radius_m = 6.371 * 10**6
earth_soi_m = 9.24 * 10**8

t = 0

x = earth_radius_m + 200 * 1609 # Initial altitude is 200 miles.
y = 0

x_vel = 0
y_vel = Math.sqrt(6.673*10**-11 * 5.98*10**24 / x)

while Math.sqrt(x**2 + y**2) < earth_soi_m
  t += 1
  vel = Math.sqrt(x_vel ** 2 + y_vel ** 2)
  r = Math.sqrt(x ** 2 + y ** 2)
  x_vel += x_vel / vel * fusion_rocket_acceleration_mss
  y_vel += y_vel / vel * fusion_rocket_acceleration_mss
  x_vel -= x / r * (earth_radius_m / r) ** 2 * 9.81
  y_vel -= y / r * (earth_radius_m / r) ** 2 * 9.81
  x += x_vel
  y += y_vel
end

puts t / 3600.0
puts vel / 1609
puts t * fusion_rocket_acceleration_mss / 1609

If you set the initial altitude to 6,000 miles (safely outside the inner Van Allen belt), the efficiency improves: you still leave Earth's SOI at 5.8 mps, but you only spend 6.6 mps delta-V, and it takes 60 hours.

Where are you getting 8.3 MPS from? Earth's escape velocity is only 6.96 and you only need 30% of that to break orbit if you're already in orbit, ~2 MPS.

[AlexanderHowl]

At 0.005g, the fusion rockets of GURPS are incapable of using the Oberth Effect to gain additional delta-v, so they have to supply all of the delta-v. The established delta-v calculations for reaching escape velocity assume an Oberth Effect, so you need to double total escape velocity delta-v when using low thrust systems (anything less than a thrust equal to the gravity of the body that a spacecraft is orbiting or escaping starts to suffer). The efficiency gains of the Oberth Effect is one of the reasons we can do spaceflight with chemical rockets.

[Michael Thayne]

Originally Posted by scc

Where are you getting 8.3 MPS from? Earth's escape velocity is only 6.96 and you only need 30% of that to break orbit if you're already in orbit, ~2 MPS.

It's not the delta-V you must burn, it's the delta-V you will in fact burn if you accelerate prograde at a constant 0.005G thrust starting in LEO until you get to the edge of Earth's sphere of influence. Not that you'd stop burning at the edge of the SOI, it's just that once you get there the calculations get a lot easier. The assumption is that you're going to be under 0.005G thrust for the entire journey. However, even if you're not planning on thrusting continuously the entire journey, it will cost more than 2 mps to break orbit using a low-thrust rocket. Basically, a low-thrust rocket prevents you from exploiting the Oberth effect for all its worth.

[Michael Thayne]

It occurs to me you could do a crude model of inner-system fusion rocket maneuvers using something similar to the tactical combat system, except hexes are 0.1 AU, and "rounds" are 6 days. On this system, starting velocity and burn points are mps / 20, and thrust rating is G * 200. The sun's "surface G thrust" is a whopping 5600 under this system, but if you pretend the sun has a 1 hex (0.1AU) radius, you can get a more reasonable "surface" G-thrust of 12. That implies that on this system, the sun's gravity is an issue if and only if you're going to pass within 0.3 AU of the sun—inside the orbit of Mercury.

[AlexanderHowl]

Well, the surface gravity of the Sun is 28g and, with a radius of 400,000 miles, that suggests that the effective attractive force (using inverse square rule) at 0.3 AU would be 0.056 m/s^2 (0.0057g).

[Michael Thayne]

Originally Posted by AlexanderHowl

Well, the surface gravity of the Sun is 28g and, with a radius of 400,000 miles, that suggests that the effective attractive force (using inverse square rule) at 0.3 AU would be 0.056 m/s^2 (0.0057g).

Yeah—it's the point where the sun's gravity is about equal to your engine's thrust. A more accurate simulation might model it when it's a significant fraction of rocket thrust, but this is probably OK if you're travelling at 60 mps or more on your way from Earth to Mars when they're roughly in opposition. Or at least I think it's probably OK?

[AlexanderHowl]

As long as you have sufficient orbital velocity, it should not matter (gravity loses only occur of you have insufficient orbital velocity). The weird thing is that you would need to speed up before you go in or slow down after you go out, as you have to account for the change in required orbital velocity. There is a reason why 'rocket science' is a synonym for 'hard'.

[Anthony]

If you have unlimited delta-V, you can cancel out gravity as long as your thrust exceeds gravity, and mostly not worry about it if you've got twice much. Problem comes when you have limited delta-V.