So here is a basic kind of relativity question to start.
For a short story I am writing, I am trying to create an close to accurate space voyage which would take just over 42 years planet time and 5.5 years ship time. So I have found this calculator which can help me figure out both the distance and the % of the speed of light at which my ship makes the journey, if they are traveling at their cruising speed.
But there is the rub. They don't start out at cruising speed and the acceleration time will dramatically change the distance traveled. I want to be able to calculate this as well. I am thinking that it might be something like a 9 month acceleration and a 9 month deceleration on the other end. But I don't remember enough math to figure out what equation I need to calculate the distance traveled during those 9 months. I can easily find an acceleration calculator online. But the distance traveled at a given rate of acceleration over a period of time I haven't yet found that. Can someone point me in the right direction?
From there I can figure the distance traveled at top speed and then add the acceleration/deceleration together with the top flight speed to get the distance traveled. I am hoping to get a number somewhere like 30 light years etc.
Are there any other factors which I have forgotten to include when thinking about this journey problem?
Oops, meant to reply to your message. Hopefully, a helpful response is below. :-)
A couple questions to get this started. Do you want the entire journey, including acceleration and deceleration, to be 5.5 years ship time, so 4 years ship-time at cruise speed?
The distance traveled while accelerating over a period of time gets into the fun of what acceleration actually means. The first derivative of position with respect to time is velocity, the second derivative (or the first derivative of velocity) is the acceleration. Easy! We have acceleration so all we have to do is work backwards through integration to position! Hopefully you are looking for uniform acceleration where the acceleration value is the same, ie, the magnitude of velocity is changing as a constant otherwise we are going to have to get into some really fun equations. We aren't going to be able to avoid calculus here as it stands. Let's start by coming up with the general set of integrals to solve. The value we really want to end with can be represented by the double integral of acceleration with respect to time over the time period of interest, represented by:
Now, what is interesting is that we are looking at equations that are dependent upon time, but your relativistic effects means time is different for the people on the ship versus the people at a rest frame or the Earth observer time. For an Earth-bound observer, where we define Earth, relatively, as a rest frame so time is constant, it is a great deal easier. Let's also assume a constant acceleration so that we have a nice clean formula for acceleration.
We now have a wonderfully simple equation defined for our acceleration. Since we know this will end up as a constant as all quantities are known, why don't we just pull the formula out and call the answer A as the acceleration constant. We next want to find our velocity as a function of time to work our way back to position as a function of time. To do this, we just integrate the acceleration over time.
We now have a general equation to determine our over a time period with constant acceleration. From here, we just integrate again over but this time over the velocity formula and we will go ahead and just integrate over the time period of interest.
We are almost home, we just need to add back in our formula for our acceleration above to replace A:
Now we can easily figure out the distance traveled while accelerating and decelerating. Actually, they are mirrors so if we figure out acceleration, we also know how far it will take to slow to 0 m/s.
Now, let me know if you would rather calculate from time being an observer on the ship or if you have an idea for a non-constant acceleration you would rather try. The distance covered doesn't matter for each case, the distance covered is the same for an observer on the Earth or on the spacecraft. It is only the amount of perceived time that will be different.
Shoot, wish there was a way to edit a post. in the last two equations, at the very end, it shouldn't be:
It should just be v_cruise. This is of course found by taking your % of the speed of light and multiplying by the speed of light, c, in meters per second.
Unfortunately all of that is using Newtonian acceleration and velocity. When you being to talk relativistic effects, there is a small correction factor that stops being ignorably small. And of course the concepts of perception on distance and time get quite muddled up as well, making many of your statements hard to claim as accurate or not (gods I love relativity, but it sure isn't an easy discussion to have when in person with ample black/white boards, let alone online...)
I really cannot explain it better than The Relativistic Rocket does. If the math is more than you are comfortable with, say what your translation is, and I can help you figure out how far off the mark you are(n't).
I assume you'll wind up having some questions about the chart if you don't follow the math, so I'll break that down a bit:.
1 year 1.19 yrs 0.56 lyrs 0.77c
2 3.75 2.90 0.97
5 83.7 82.7 0.99993
8 1,840 1,839 0.9999998
12 113,243 113,242 0.99999999996
Using the second line of their chart:
Accelerating at 1g the entire time, the crew will age 2 year, while their family at home will age 3.75 years. This will put them 2.90 Light Years away from Earth, and they will currently be traveling at 0.97% the speed of light.
Since you want to have them arrive at the destination, not blow past it, you have to double all the numbers involved. So you know that you will be going much further than 2.9 Light Years for your situation, and getting closer to the speed of light in the process.
One major factor you will have to account for in your book is communication time, which is so often ignored when people talk about these things. Information is also unable to break the speed of light,
but you can cheat a little and claim to be able to match it (which is still unrealistic, but more readily believable, and roughly appropriate).
So in the doubled case from above, 4 Years of Crew time, 7.5 Years of Earth time, and 6.8 Years for a message to transmit from destination back to earth, the crew won't say "We got here!" to Earth until 14.3 Years (Earth Time) after they leave, and they won't get a "Congratulations!" response from Earth until 17.6 years (traveler time) after they left.
Go find and watch the Anime short "Voices of a Distant Star." This has a young girl flying off to fight some aliens far away and texting with her boyfriend back home. He ages and the time between messages grows longer and longer as she gets further away. --Note, this anime only deals with the time to communicate at long distances, not with the time shifting effects of travel. Pretty sure I remember them copping out and using Wormholes or something like that for the 8 light year jump.