At 01:08 PM 12/19/02 -0600, Ronn! Blankenship wrote:
At 04:33 AM 12/19/02 -0500, Damon wrote:Hi all,
Thanks for the responses so far...
No, I don;t need relativistic flight times. Perhaps it would be helpful if I explain the problem.
In my campaign a ship just "jumped" into system, using a jumpgate. The jumpgate orbits the system's primary at around 100AU. The planet the ship wants to go to is around 2 AU. What is the formula to calculate the time it would take (non-relativistically) for the ship to reach its destination using the acceleration/deceleration method?
Hi, Damon,
Again, if by "the acceleration/deceleration method" you mean that the ship's engine is capable of generating a constant thrust for an indefinitely long period of time, so you can accelerate at a constant rate from the start to the midpoint, then turn around and decelerate at the same rate from the midpoint to the destination, then, over the kind of distances you are talking about, if the acceleration is even a reasonable fraction of 1g, you will rapidly build up such speed that to a good first approximation the ship will be moving in a straight line, so you can use the formula s = 1/2*a*t^2. In this case, let s = 49 AU, and a be the acceleration of the ship, and you will get the time to the midpoint, and the total time will be twice that. (This also assumes that the ship has some kind of super-duper engine which can do the above on an amount of fuel which is negligible compared to the mass of the ship. Otherwise, one has to take into account that the mass of the ship is constantly changing as fuel is consumed, and the problem becomes more complicated.)
By contrast, spaceships using chemical rockets burn a lot of fuel very fast to create a high acceleration which lasts only a short time, then coast the rest of the way to the destination, at which time they must again burn fuel to slow down. To a first approximation, the acceleration or deceleration phase of such spaceships can be considered instantaneous. In between, the ship follows an orbit just like an asteroid or comet, and the same equations may be used to describe its orbit. To a first approximation, the lowest energy (= least fuel consumption) orbit is an arc of the ellipse which is tangent to the orbits of the starting planet (or other body) and the destination planet. To be more accurate, you would need to take into account the gravitational pull of all the other bodies in the system and how they affect the path of the spaceship.
HTH.
Now I must be off to pick up finals and term papers to grade. Oh, joy.
I meant to add this earlier but was in a hurry to leave:
If I've understood your question correctly, in the situation you're describing, one could almost give the infamous accountant's answer: "What do you want the answer to be?" I.e., if you assume the ship has a variable acceleration, or that it goes by an indirect route, you can make the time required pretty much fit anything you need for the plot of the story or game. The formula I gave you above will give you the absolute minimum time (which in many cases may be what you want), if you use the maximum acceleration of which the ship is capable.
Frex, assuming a = 1 Earth gravity = 9.8 m*s^-2,
s = 1/2*a*t^2
49*1.495e11 = 1/2*9.8*t^2
[1 AU = 1.495e11 m]
49*1.49e11/4.9 = t^2
t = sqrt (1.49e12)
t = 1.22e6 seconds
t = 14.13 days
So it would take about 4 weeks (2 weeks of acceleration at 1g followed by 2 weeks of deceleration at 1g). Is that good enough for what you need?
-- Ronn! :)
Ronn Blankenship
Instructor of Astronomy/Planetary Science
University of Montevallo
Montevallo, AL
Disclaimer: Unless specifically stated otherwise, any opinions contained herein are the personal opinions of the author and do not represent the official position of the University of Montevallo.
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