[Spaceships] Travel Time at Relativistic Velocities

munin · 09-06-2010, 10:24 PM

The formulas used to calculate STL travel times in GURPS Spaceships seem to produce incorrect results when velocities are large enough for relativistic effects to become significant. For example, they might give travel times (in years) shorter than the distances (in light-years). The only mention of relativistic effects I can find is a quick mention of time dilation (GURPS Spaceships 5, p. 11). I run into this problem mostly with superscience engines, but even a ship with Antimatter Pion engines can reach a significant fraction of light-speed.

Online, I found an equation for calculating the time required (WRT non-accelerating frame of reference) to travel a specific distance at a constant acceleration, with no derivation to check but which produces sensible results (d=distance, c=light-speed, and a=acceleration -- in consistent units, not GURPS Spaceships units):

(1) t = [(d/c)^2 + 2*d/a]^(1/2).

In English, that boils down to saying that the time required will be equal to the time required by light (d/c), plus some time inversely proportional to your acceleration, which seems reasonable.

If you want v=0 at your destination (accelerate half-way, then decelerate the rest), the formula would be:

(2a) t = 2*[((d/2)/c)^2 + 2*(d/2)/a]^(1/2), which simplifies to:
(2b) t = [(d/c)^2 + 4*d/a]^(1/2).

That's helpful for reactionless engines, and for reaction engines when your delta-V is enough to burn the entire way.

But things get more complicated when you don't have enough delta-V to burn the whole way, and have to spend some of the time coasting. In that case, you have a certain amount of delta-V you can spend, but the velocity you end up with (WRT non-acc) will be less than the delta-V spent -- so how do you calculate the time required to spend that delta-V, and determine how far that got you? I'm not even sure how to determine whether you have enough delta-V to burn the whole way. The equation given for v on the online page is for v WRT a non-accelerating ref frame (i.e., less than c), whereas your delta-V might give you an apparent v > c (right?), so I don't think I can simply use the rocket equations there to solve for d or t in terms of v (i.e., delta-V).

So my questions are:

Are the above equations, (1) and (2b), the correct equations to use to determine travel time with no coasting?
How would you use those equations (or others) to determine a travel time (and the distance covered by acceleration) when you accelerate for some delta-V (not for some d or t), coast at a relativistic velocity, and then decelerate to v=0?
Probably related to #2, how do you determine whether you have enough delta-V to burn all the way, or whether you have to do some coasting?

Thanks!

Langy · 09-07-2010, 12:23 AM

I made an excel spreadsheet a while back for travel time calculation.

Here is the file

It includes delta-V calculation (based on ship-board burn time and the acceleration of the ship), the ability to specify different initial and final burns, and calculating figures for either coasting or constant acceleration, along with objective and ship-board travel time calculations. It doesn't, currently, let you specify an initial velocity, though, and I'd need to figure some stuff out in order to get that plugged in. Sorry.

To get at it, open that file up (it's an excel 2007 file) and go to the Relativistic Travel sheet. There's also calculations for non-relativistic travel set up, which are slightly different (they take into account the targets escape velocity, if it has one, allowing you to go into orbit at the end, for example).

09-06-2010, 10:24 PM	#1
munin Join Date: Aug 2007 Location: Vermont, USA	[Spaceships] Travel Time at Relativistic Velocities The formulas used to calculate STL travel times in GURPS Spaceships seem to produce incorrect results when velocities are large enough for relativistic effects to become significant. For example, they might give travel times (in years) shorter than the distances (in light-years). The only mention of relativistic effects I can find is a quick mention of time dilation (GURPS Spaceships 5, p. 11). I run into this problem mostly with superscience engines, but even a ship with Antimatter Pion engines can reach a significant fraction of light-speed. Online, I found an equation for calculating the time required (WRT non-accelerating frame of reference) to travel a specific distance at a constant acceleration, with no derivation to check but which produces sensible results (d=distance, c=light-speed, and a=acceleration -- in consistent units, not GURPS Spaceships units): (1) t = [(d/c)^2 + 2d/a]^(1/2). In English, that boils down to saying that the time required will be equal to the time required by light (d/c), plus some time inversely proportional to your acceleration, which seems reasonable. If you want v=0 at your destination (accelerate half-way, then decelerate the rest), the formula would be: (2a) t = 2[((d/2)/c)^2 + 2(d/2)/a]^(1/2), which simplifies to: (2b) t = [(d/c)^2 + 4d/a]^(1/2). That's helpful for reactionless engines, and for reaction engines when your delta-V is enough to burn the entire way. But things get more complicated when you don't have enough delta-V to burn the whole way, and have to spend some of the time coasting. In that case, you have a certain amount of delta-V you can spend, but the velocity you end up with (WRT non-acc) will be less than the delta-V spent -- so how do you calculate the time required to spend that delta-V, and determine how far that got you? I'm not even sure how to determine whether you have enough delta-V to burn the whole way. The equation given for v on the online page is for v WRT a non-accelerating ref frame (i.e., less than c), whereas your delta-V might give you an apparent v > c (right?), so I don't think I can simply use the rocket equations there to solve for d or t in terms of v (i.e., delta-V). So my questions are: Are the above equations, (1) and (2b), the correct equations to use to determine travel time with no coasting? How would you use those equations (or others) to determine a travel time (and the distance covered by acceleration) when you accelerate for some delta-V (not for some d or t), coast at a relativistic velocity, and then decelerate to v=0? Probably related to #2, how do you determine whether you have enough delta-V to burn all the way, or whether you have to do some coasting? Thanks!

09-07-2010, 12:23 AM	#2
Langy Join Date: May 2008 Location: CA	Re: [Spaceships] Travel Time at Relativistic Velocities I made an excel spreadsheet a while back for travel time calculation. Here is the file It includes delta-V calculation (based on ship-board burn time and the acceleration of the ship), the ability to specify different initial and final burns, and calculating figures for either coasting or constant acceleration, along with objective and ship-board travel time calculations. It doesn't, currently, let you specify an initial velocity, though, and I'd need to figure some stuff out in order to get that plugged in. Sorry. To get at it, open that file up (it's an excel 2007 file) and go to the Relativistic Travel sheet. There's also calculations for non-relativistic travel set up, which are slightly different (they take into account the targets escape velocity, if it has one, allowing you to go into orbit at the end, for example).