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This question is really about best possible travel times between celestial bodies. In my current sci-fi setting, the Human race is spaceborne, but incapable of FTL travel. Their thrusters are excellent, fuel is not an object; the real limiters on travel times will therefore be issues (if I'm right) of acceleration - acceleration is bad for us unless its taken in moderation, so speed-up and slow-down times will have to be kept at safe limits; as far as I know, actually going really fast is not, itself, dangerous - as long as you don't stop suddenly.

A little exposition. Humanity is some 4000 years off of Earth, but without FTL, it is largely bound to the solar system, where it resides in thousands of more-or-less self-sustaining habitats and orbitals (Earth is held in preserve by the Ruling Family - think "the King's Forest" but with a whole planet). Several hundred years after we left Earth, constructor ships were sent to Alpha Centauri equipped to construct similar habitats there. A parallel civilization exists around that star, roughly 4.5 lightyears away, one of my biggest questions is how long it takes for a trip from Sol to Alpha Centauri? Even with my rudimentary understanding of the Big Physics involved here, it seems highly unlikely that either group could exert any reasonable political influence over the other, as any threat of force would be at least decades away.
- However, while intra-system flights would likely have flight times that were feasible with 1 Earth-G (our most comfortable level) worth of acceleration, flights to Alpha Centauri would likely want to use much faster acceleration - I know we sometimes put people past 1G (Fighter Pilots for instance, or - go figure - Astronauts), I'm uncertain, however, what kind of effect that would have on the crew after sustained exposure. In any case, suspension or hibernation of one form of another would be necessary for this trip to be feasible; could such a mechanism alleviate the risks of high-G exposure, and if so, how fast could you reasonably accelerate a Human before things started to get... dicey?

More simply, what kind of travel times am I looking at for planet-to-planet intra-system flights, and star-to-star flights utilizing hibernation or suspension (with greater acceleration allowances, acceleration times could be traded for time flying at peak velocity), considering that thrust and fuel are not an issue?

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    $\begingroup$ There seams to be an answer to almost this exact question over at space exploration SE:space.stackexchange.com/questions/840/… $\endgroup$ – Josh King Mar 9 '17 at 16:55
  • $\begingroup$ You're right! There's a really good description of the mathematical crunch needed to describe sub-light travel there - I can definitely make use of it. That said, this question is partially about feasible acceleration and speed - I'm still concerned about my crew, and unsure how fast I can make them go (and how fast I can get them to "fast") safely. $\endgroup$ – CelestialCaveBear Mar 9 '17 at 17:12
  • $\begingroup$ There are some problems with higher speeds: space isn't completely empty. Within the solar system, things get better until you reach 900 km/s, the speed of the solar wind, thereafter it gets worse — but with the extra disadvantage that you're catching up with the radiation and therefore need a radiation shield in the front not just behind where your engines are. At 9900km/s relative to Earth (0.033c) you get x10 the particle count, each at x10 the relative velocity. $\endgroup$ – BenRW Mar 9 '17 at 17:17
  • $\begingroup$ Hibernation is not necessary as people can experience time dilation, when acceleration and reactive mass are not a problem. You might take look at the Q/A physics.stackexchange.com/questions/75002/… $\endgroup$ – MolbOrg Mar 10 '17 at 2:31
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    $\begingroup$ Your headline talks about speed, but the body of the question talks about acceleration. Which of those do you mean? $\endgroup$ – Mike Scott Sep 3 '17 at 16:02
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At 1g you are flirting with light speed in a year. So from an external reference frame a reasonable first approximation is x + 2 where x is the distance in light years. Externally there's not much benefit after 90% of light speed. Internally of course it takes less ship time the faster you go.

You have to deal with the whole problem of interstellar hydrogen. At c, you are doing 300,000,000 m/sec which at a density of 1 atom/cm3 results in a flux of 300,000,000,000,000 or 3e15 atoms per square meter per second hitting the hull.

That's roughly 2e-9 kg. But mv2/2 = 3e8 * 3e8 * 2e-9 / 2 = 9e7 Joules/sec

Or about 100 megawatts/square meter of cross section. Cooling is going to be interesting. this may be the speed limiting factor. Remember this is the density of neutral hydrogen. Not going to deflect with a magnetic field. And we're ignoring the Einsteinian Elephant in the room.

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  • $\begingroup$ Well - I handle objects hitting the hull with the setting's eponymous phlebotinum, Hard Light, in this case Hard Light Shields used to create a wedge which protects the ship from faceward impacts, although there would likely still be a practical limit on travel speed (how much the shield could deflect). I'd like to know about this Elephant, though - are you referring to Time Dilation? I'm working on time dilation. $\endgroup$ – CelestialCaveBear Mar 10 '17 at 14:34
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Since $t = \sqrt{\cfrac{2d}{a}}$ you can see that doubling your acceleration only decreases the time by the square root of 2. Halving the time requires quadrupling the acceleration.

$$ d = 100, a = 1 \Rightarrow t = \sqrt{\cfrac{200}{1}} = \sqrt{200} \approx 14.14 $$

$$ d = 100, a = 2 \Rightarrow t = \sqrt{\cfrac{200}{2}} = \sqrt{100} = 10 $$

$$ d = 100, a = 4 \Rightarrow t = \sqrt{\cfrac{200}{4}} = \sqrt{50} \approx 7.07 $$

So you can calculate how long it will take to get to where you're going at a safe 1 g, and decide how fast you want your ships to get there, and come up with what level of sustained acceleration your sci-fi civilization has to be able to do.

For example, let's say you wanted to get from Earth to Trappist 1, that system with all the possibly habitable planets. That's $3.7 \times 10^{17}$ m away. We can plug that into our equation, BUT they need to decelerate before they get there. So accelerate for half the journey, decelerate for half the journey. Decelerating for half the journey is the same as accelerating for half the journey. So we need to calculate the time to get to the halfway point twice.

$$ d = \frac{3.7 \times 10^{17} ~\text{m}}{2} = 1.85 \times 10^{17} ~\text{m} \\ t = 2 \sqrt{\cfrac{2d}{a}} $$

Plug in $a = 1~\text{g} = 9.8~\text{m/s}^2$ and we get $3.89 \times 10^8$ seconds or 12 years.

Not fast enough? Maybe your scientists develop special pills to resist the g forces, I dunno. Now they can sustain 2g. That'll be $2.75 \times 10^8$ seconds or 8.7 years or $\sqrt{2}$ less than before.

Maybe they develop acceleration couches and advanced sleep tech so they can sleep the whole way at 4g. That doubles the acceleration, so 6.15 years. Quadruple the acceleration to half the time.

In this way you can calculate the time once and then estimate the impact of changing the acceleration.

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    $\begingroup$ Your numbers would work for a short interplanetary trip, but are off, if going to interstellar distances you could never accelerate to a velocity higher than the speed of light. You would accelerate mostly as you said and hit a max speed and then decelerate in the same process. Trappist 1 is 39.5 Light Years away, it would take a minimum of 39.5 years to get there if you could accelerate instantly, so it would take more than that as it takes time to accelerate up to c. $\endgroup$ – Josh King Mar 9 '17 at 22:13
  • $\begingroup$ @JoshKing You're totally right! Derp! Hmm... I wonder what the acceleration equations are then. $\endgroup$ – Schwern Mar 9 '17 at 22:15
  • $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$ – MolbOrg Mar 10 '17 at 2:27
  • $\begingroup$ or better this and that answers $\endgroup$ – MolbOrg Mar 10 '17 at 2:38
  • $\begingroup$ and to follow up on Josh's comment, the time at close to light speed would be 39.5 years minimum as viewed by people remaining at the launch site (so ~80 years minimum for these people to get back any message of success for the mission). However, the crew members on the ship could well experience it as only 12 years due to relativistic effects (I cannot recall enough of the maths for this to remember whether the numbers come out in the same scale is if you'd ignored relativity - probably not, although with the numbers in this answer it may well be close). $\endgroup$ – Neil Slater Jul 15 '17 at 13:44
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A few calculations:

Assuming that you're going to travel at, say, 0.1C it will take you around 40 years to reach Alpha Centauri, and only around a month to accelerate to that speed at 1G.

If you can hit 0.5C then it becomes a bit more relevant; ~8 years to travel the distance and six months to accelerate.

(Obviously you'll need to accelerate at both ends, but handily the time taken to accelerate actually works out as the total extra time if you do both ends)

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  • $\begingroup$ There's no reason you couldn't actually reach C, though, in the Alpha Centauri flight (well, very close to C, assuming no massive object can properly achieve C). Correct? $\endgroup$ – CelestialCaveBear Mar 9 '17 at 17:24
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    $\begingroup$ 8 years? I get much closer to 7 years when I apply the Lorentz factor. $\endgroup$ – Aron Mar 10 '17 at 6:49
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You're right that it's not about speed per se but about acceleration & deceleration the sustained limit there is about 2g if you want to transport people and not puddles of goo, 1g would be much more comfortable though. I've never checked the math on these but Larry Niven is usually pretty good; so at 1g with a rollover to decelerate halfway, also at 1g it's 4 years to Centuri, 21 to the centre of our galaxy and 28 to Andromeda. These are ship times, the outside universe will experience more time thanks to general relativity. I think at 2g you could knock about a quarter off the transit to Centuri but the other two wouldn't change all that much because they hit and sustain the maximum fraction of C for the drive system for much of the trip.

Incidentally the main limiter on total velocity with a reaction drive is the speed of your exhaust; at some point it will be leaving the vessel as fast as the vessel would be leaving it behind if it wasn't moving at all, some time a long while before this it becomes un-economic to run your engines anyway because the speed gained is so much less than the energy cost of getting it, if you're talking a Bussard Ramjet (THE go-to when you're talking about sourcing fuel en route) the point of diminished return and the point of non-acceleration are both pretty close to C anyway, which is why they're so popular as a slower-than-light engine.

Edit: Sorry just saw an article on the Bassard design, it doesn't work, at all.

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