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I'm about to run a sci-fi campaign with sublight starship travel, with a solar system and to all the stars within 10 light years of Sol.

There are starships that accelerate at 1g, and some that accelerate at 2g, and they can change the acceleration rate a bit for comfort.

Given that I have variable acceleration rates, and variable distances, I need: A formula for how long the passengers think it takes to travel X parsecs under Y acceleration. A formula for how long an outsider thinks it takes the passengers to travel X parsecs under Y acceleration.

closed as off-topic by user14111, phantom42, Izkata, The Fallen, Joe L. Oct 1 '14 at 1:35

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  • So, you want to incorporate relativistic time dilation? – user1027 Oct 1 '14 at 1:06
  • There's a lot of theoretical info on what happens as an object accelerates to near-lightspeed, but I've never heard a good explanation for what happens when a relatavistic object slows down. – Joe L. Oct 1 '14 at 1:31
  • See How long would it take me to travel to a distant star? on the Physics SE – John Rennie Oct 1 '14 at 8:24
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This seems more like a physics question that a science fiction question, the physics stack exchange would be a better choice for issues like this. But for this question, I think the relativistic rocket page has everything you need, if the ship has some constant acceleration a the formulas can give you the distance traveled d or velocity v as a function of time t, all measured in the frame where the ship started to accelerate from rest, as well as the onboard time T (different than t because of relativistic time dilation). Some of the formulas involve hyperbolic trig functions, but you can calculate them with the online calculator here. Just make sure you're consistent with your units--the formulas become simpler if you use years for time and light years for distance, since in those units c=1 (and as noted on the page, 1G acceleration corresponds to a = 1.03 light-years/year^2, similarly 2G would be a=2.06 and so forth). If you want to figure out distances in parsecs, you can just calculate them in light years and multiply by 0.3066 to get the corresponding distances in parsecs.

I should also add that the formulas are to continuously accelerate to your destination, meaning you will reach it at high velocity--but if you want to imagine accelerating half the way and decelerating the second half, just figure out the time to travel half the distance and double it. Changing accelerations midway through the trip would be more complicated, when the acceleration changes you'd probably need to switch to a new frame where the ship is at rest at the moment of the change and then for the second leg of the trip you could use the same formulas but with d and v and t referring to the new frame--to relate the coordinates of different frames you need to use the Lorentz transformation, though only the simplest version given in the "boost in the x-direction" section should be required.

One other issue you may want to consider is the ratio of fuel mass to payload mass, which will depend on what type of propulsion system is used--see the "relativistic rocket equation" section of the relativistic rocket wiki page for some details.

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