# Math of relativistic space travel

1. Mar 26, 2009

### Ike47

Hello. This is my first post here, so I can only hope (having read the guidelines and physics FAQ), that my post is appropriate to go here. If not, and it is deleted, I'd much appreciate an explanation by PM and a suggestion as to where it would be appropriate to post.

That said, the reason for my questions is that I am in the preliminary stages of try to write a science fiction novel. I'm 62 and have not written previously, although I have a lifelong interest in science, especially astronomy, cosmology, and particle physics, and I've read many hundreds of of SF (and other) novels. The idea of starting to write at my age (now that I'm retired) is probably quixotic, but I'd still appreciate answers to my questions for their own sake, even if I never write a paragraph.

If anyone's still reading, my request is for the mathematical formulas to calculate how long it would take to travel a set distance (in ly or parsecs), both from the traveler's reference frame and from the reference frame of someone remaining at the point of origin of the trip. I would like the formulas to be able to handle not only constant acceleration (and deceleration), but also acceleration for part of the trip, then 'coasting', then deceleration for the final part of the trip.

Of course, the formulas won't be of much help to me if they are too complex to calculate without powerful computer assistance, so it may be necessary to simplify the formulas so as to be practical to use and only give approximately accurate results. For example, it may not be feasible to include in the formulas the steadily changing mass of the vehicle as fuel is expended.

It would also help to have access to some table of the energy per mass unit that different types of fuel can produce, such as chemical (lox, etc.), fusion materials, and fission materials. I guess anti-matter materials would be interesting as well, although I don't see that as a feasible fuel. And I guess such a chart should also show the maximum efficiency likely with each type of fuel (thrust-producing energy/total energy output).

I have read a number of books that cover, in varying degrees, the topic of relativistic space travel, but none that I've come across are very specific in this particular area. Examples are books by Peebles, Thorne, Barrow and Tipler, Davies, Hawking, Maffei, Smoot and Davidson, Kaku, and Weinberg. Of course, if there is a book I haven't come across (of which there are definitely many) which covers the area of my interest in detail, I'm much appreciate knowing that.

I've also tried looking online, but I've been able to find only two items that are at all useful. The lesser one is a page called 'Space Math' on the site cthreepo.com, which gives a java script for calculating time or distance or acceleration if two of the three variables are given. However, the basic formulas upon which the script is based are not given, nor does the calculation allow for 'coasting'. It assumes (constant) acceleration for half the journey, then (constant) deceleration for the other half.

The more useful page is one by Philip Gibbs, updated by Don Koks, called 'The Relatavistic Rocket' (http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html [Broken]). This does in fact give a number of very useful formulas, but its shortcomings include the assumption of constant acceleration/deceleration (no coasting), and idealized fuel and efficiency.

It doesn't allow one to calculate, for example, given a maximum feasible amount of fusionable fuel, how long it would take (by the traveler's and by Earth's clock) to travel, via acceleration, coasting, and deceleration, approximately 4ly (Alpha Centauri) or 12ly (Tau Ceti) or 20ly (82 Eridani).

Now that I think about it, in the previous example, I guess there is also the question, if fuel is limited, how quickly (half of) it should be consumed to produce the minimum total trip time.

In any case, I've rambled on far too long as it is. Any answers, recommended sources, or corrections to my questions would be much appreciated.

Last edited by a moderator: May 4, 2017
2. Mar 26, 2009

### DaveC426913

Hm. I keep needing this too.

I calculated http://www.davesbrain.ca/science/gliese/index.html" [Broken] a while back. I should really generalize that algorithm and post it somewhere.

BTW, I make use of http://www.1728.com/reltivty.htm" [Broken] a lot. Skip to the bottom.

Last edited by a moderator: May 4, 2017
3. Mar 26, 2009

### Ike47

Thank you, Dave.

The Gliese 581 page is certainly enjoyable to read, and all the more so because of Gliese 581c. Still, the numbers crunched for the time a trip there would take use the standard (and to me, unrealistic) assumptions of 1g acceleration and deceleration for the whole trip. What I'd like to figure out is if one used the best currently conceivable method of space travel, which I presume to be a fusion engine (since antimatter would be virtually impossible to amass in sufficient quantities and a 'scoop' engine has apparently fatal flaws), just how much acceleration could a fusion engine produce, and for how long.

For example, perhaps the maximum acceleration a fusion engine, using a readily available and low-mass fuel, could achieve might be .5g. And it might be that it could only sustain that acceleration for 1 year (ship time), allowing for a similar deceleration in the last year of the flight, because the amount of fuel mass needed to sustain the acceleration longer than that would lower the maximum acceleration below .5g. So, the question would then be, using these purely made-up numbers, how long would it take for a vehicle to travel approximately 20ly if it accelerated at .5g for one (ship reference) year, then coasted, then decelerated at .5g for the last (ship reference) year? Of course, there would be two answers, one for the ship and it occupants, another for the time taken from Earth's perspective.

As for the relativity calculator, I took a look at that too, and I understand what it does itself, but I'm not sure how to use its results to measure relativistic travel.

4. Mar 26, 2009

### DaveC426913

1g is easily doable. Rocket engines do 8gs or more.

The problem, as you know, is really that there is no conceivable method of space travel that can meet the fuel requirements. No matter what the method, there's just no way you can bring that much fuel. Antimatter is by far your best bet - much better than fusion - but still... Someone did a calc of the fuel requirement here previously. It was still a very high fuel-payload ratio.

The inescapable conclusion then, is that we must postulate a heretofore undiscovered drive mechanism whose fuel requirements are more practical.

In a nutshell, once you somehow make the trip feasible AT ALL, there's no reason NOT to go at 1g. (Not the least of which , the cumulative effect of relatavistic time dilation makes the trip MUCH shorter.)

Last edited: Mar 26, 2009
5. Mar 26, 2009

### DaveC426913

So how do you propose to bring enough fuel for this fusion motor? Since it's going to be a trip of decades or more, you're going to be bringing a pretty big payload. And since your fuel-payload ratio is ridiculously high, then you're going to be carrying (and pushing) astronomical amounts of fuel. You're going to need to go in a small asteroid...

6. Mar 26, 2009

### Naty1

For variable speeds and accelerations you will need some compuer support...why make it so difficult...it's not that an interesting part of of a typical fiction story...just don't bee too specific. Star Wars did well without such detail!

Or, in the first Chapter of Black Holes and Time Warps, Kip Thorne gives some calculations for several different trips...use his numbers if you like; nobody will ever know the difference especially if your use fictictious origins and destinations....

7. Mar 26, 2009

### DaveC426913

That's because Star Wars is Fantasy, not Science-Fiction.

The OP is obviously writing a story in the non-fictional and non-fantastic stellar neighborhood. Trip times will matter. So will plausible science.

8. Mar 26, 2009

### DaveC426913

I just did a search on PF for "antimatter fuel payload" and turned up a half-dozen threads from recent months/years discussing trips through our local stellar neighborhood.

9. Mar 26, 2009

### DaveC426913

Last edited by a moderator: Apr 24, 2017
10. Mar 26, 2009

### Ike47

Let me respond to at least a couple of points. Yes, I'd like to write a novel that takes place in the non-fictional 'local area' of space, starting with a radius from our solar system of perhaps 20 to 50 ly. I don't expect it to be 'realistic', but rather 'plausible'.

Let's take an example. In The Anthropic Cosmological Principle, Barrow and Tipler, assume that any species attempting interstellar communications (including us) will be able to produce "von Neumann probes" (self-replicating universal constructors) in 100 years. Personally I find this preposterous. They even posit that such constructors could create human beings 'from scratch', by creating human DNA from the simple elements it is made of. I consider such speculation to be not plausible.

On the other hand, if, 500 years from now, mankind still has not achieved an anti-matter engine, or possibly even a practical fusion engine, would any human beings have the will and wealth to begin a trip to Alpha Centauri (especially if a binary/trinary can indeed have habitable planets, as assumed by Asimov, e.g., in Foundation and Earth; and if we have been able to discover such planets there telescopically from Earth) using just a fission engine and requiring multiple generations to get there? I certainly don't know, so I wouldn't say it's 'realistic'; but given the frustration of at least some at the difficulty of space exploration, I would consider it 'plausible' that at some point in the future, under these conditions, such a trip would be attempted.

quoting DaveC426913:

So how do you propose to bring enough fuel for this fusion motor? Since it's going to be a trip of decades or more, you're going to be bringing a pretty big payload. And since your fuel-payload ratio is ridiculously high, then you're going to be carrying (and pushing) astronomical amounts of fuel. You're going to need to go in a small asteroid..."

Well, this is part of what I want to find out mathematically. I'm not sure that 'an asteroid's worth' of fuel would help that much, since the increased mass may be self-defeating. But let's say we construct a fusion (or fission, for that matter) engine vehicle, and give it just enough fuel initially to park it in a orbit outside of most of earth's gravity well. Then we send up in modest increments enough fission or fusion fuel (the latter presumably having much less mass, as well as more energy production potential) to completely fuel up the vehicle with some 'practical' fuel-to-payload ratio (10:1? 50:1??).

If we do all this, and then use about 1/2 the fuel (depending on relativistic effects on mass, if any) to accelerate at 1g, then coast until it's time to use the rest of the fuel to decelerate at 1g, how long would the trip take to A Centauri? or to Gliese 581 for that matter? 50 years? 200 years? 10,000 years? These are the sort of answers I'd like to be able to calculate within, say, 20-30% accuracy.

I'm hoping someone here can tell me how to calculate this, or, if necessary, tell me it's not practicable to calculate.

I don't think I responded to all the comments in this thread, but this is a start anyway. Thanks to each of you for respodning.

11. Mar 26, 2009

### DaveC426913

No quite self-defeating, no. As you scale up your ship, there will come a point where you have enough fuel to move it. On the graph or increasing ship scale, the fuel increase will rise as a diagonal line, but your ship's mass will rise at a smaller rate, because you're not increasing engine, payload, etc., So at some point, your fuel exceeds your need. But pessimistically, that intersection may be somewhere up at the size of a small asteroid...

Nope. You'll use a very large fraction of your fuel on the outbound leg beacuse you need to haul all that fuel with you.

Say you have a ship with a payload that's 10T and it needs 100T of fuel (10:1) to get from 0 to crusing speed (whatever that is, it doesn't matter) and then back to zero. That is your homebound ship. It must start for home with a mass of 110T (payload + fuel for trip home).

Now for the outbound journey. We just established that, for every 10T of ship, we'll need 100T of fuel. And your homebound ship masses 110T, which means the outbound ship will need (110*10)T = 1100T of fuel.

i.e. Your ship starts from Earth at 1110T, burns 1000T of fuel on the outbound journey but only 100T on the homebound journey. i.e. 90% of your fuel is burned on the outbound leg.

And that's only 10:1. At 100:1 your 10T payload will require 1,000,000T of fuel. (Your 3m diameter lifepod is now sitting on a tank that is 100m in diameter)

No, the only practical way to do this trip is to manufacture your fuel at the destination. Now you're back to needing only a 110T ship.

Note though, that a 10T payload won't make it to high Earth orbit, let alone another star, so scale it up a several orders of magnitude.

The amount of fuel does not play in to the duration of the trip. The efficency of the drive and its accleration is the major factor here. And here we can only guess.

This is why - in my opinion - the acceleration will technically be as large as we need, and so the limiting factor on it will be the occupants.

Again, it is a very efficent use of resources to make the journey as short as possible (longer journey = more resources to keep occupants alive and sane, and more need to pack even longer-term consumables like rare nutrients, water, oxygen, etc.)

Last edited: Mar 26, 2009
12. Mar 26, 2009

### Ike47

13. Mar 26, 2009

### Janus

Staff Emeritus
Here's some formulas that might help:

$$t= \frac{c}{a}\sinh \left ( \frac{V_e}{c} \ln(MR) \right )$$

$$d= \frac{c^2}{a} \left ( \cosh \left ( \frac{V_e}{c} \ln(MR) \right ) -1 \right )$$

$$v = c \tanh \left ( \frac{V_e}{c} \ln(MR) \right )$$

where

t is the time
d in the distance traveled
v is the velocity attained
a is the acceleration
c is the speed of light
Ve is the exhaust velocity of your rocket
MR is the mass ratio (mass of ship+ mass of fuel)/ mass of ship.

Example: a fusion rocket might get an exhaust velocity of 7% of c
Assuming a 1g acceleration and a mass ratio of 11

t= 5162509. sec = 60 days
d= 0.0014 ly
v = 0.167c

The time dilation factor at 0.167c is 1.01 (Not much of a difference)

So, to get to Alpha Centauri, without stopping upon arrival, takes 60 days + (4.3-.0014)/0.167/1.01 = 25 yrs and 10 mo.

Stopping there takes 120 days + (4.3-.0028)/0.167/1.01 = just a tad under 26 years.

Of course stopping there takes more fuel. So if we assume that the mass ratio of 11 is for braking at the end, then the the mass ratio at the beginning of the trip is 111.

So, a trip to the nearest star with a fusion rocket and a fuel mass 110 times that of the ship and would take 26 years

14. Mar 26, 2009

### DaveC426913

Oh, I was simply using 10T as an arbitrary unit of payload for ease of calcs. A 10T payload (people, equipment,air, etc.) is way too small for any practical mission, even a short one. The payload for the Apollo missions was way more than that. For an interstellar journey, you're looking at a vehicle much bigger than any oceanliner, even before considering fuel.

15. Mar 27, 2009

### Ike47

Thank you. Alas, these equations still do not let me calculate time or distance for any journey that involves accelerating and decelerating for the beginning and end of the trip, with 'coasting' in the middle, since:

the formulae for t and d have a in the denominator, and since a is 0 during the period of coasting, the value of the function is either infinite (which is not correct, since the vehicle is still moving) or indeterminate, which obviously doesn't help; and

the formula for v has Ve (sorry, I don't know how to do subscripts, etc., here) in the numerator of tanh, giving Ve = c tanh (0) = c (0) = 0. Again, this is not 'correct', since the vehicle is moving.

Of course, all the formulae are correct if there is no acceleration at all: t is infinite, d is 0 (the fomula gives the indeterminate (c^2 / 0) x 0), and Ve is 0.

Any thoughts on how I could calculate the time or distance in a situation that includes a period of coasting (where the constant v might or might not be relativistic)?

16. Mar 27, 2009

### DaveC426913

I'm going to try to build a calculator but I can't say how long it'll take me. Keep tuned in.

17. Mar 27, 2009

### Ike47

Thank you!

I am curious as to exactly what you mean by a 'calculator'. Are you referring to determining what the necessary formulas would be? Or do you refer to an actual calculating program, where one simply inputs numbers, and the answer is produced by the program? If the former, that's great, and I'd be happy to work from there. If the latter, that would of course be even nicer, but it seems to more work than I have any right to request of you. Also, if the latter, could you share the formulas (or whatever) you would use in the calculator?

Again, thank you very much.

18. Mar 27, 2009

### Janus

Staff Emeritus

There is no one formula that will give you the answer, so you do it in sections.

For instance:

1. Use the given formulas to determine the time, distance and final velocity of the acceleration leg of the trip.

2. Double the distance and time from step 1 to get the combined totals for both the acceleration leg and deceleration leg.

3. Subtract the distance value you got in step 2 form the total distance you want to travel. This will give you the distance traveled during the coasting leg.

4. Divide the result of step 3 by the velocity you got in step 1 to get the time spent during the coasting leg.

5. Use the time dilation formula to determine the time dilation factor for the coasting leg. Apply the factor to step 4's result.

6. Add the times from steps 2 and 5 to get the total trip time.

19. Mar 27, 2009

### DaveC426913

Yes, my plan is to build a calculator. It's for my own enjoyment. I realize that this is more than you need. I also realize I don't yet know what the formulae are, and that I'll need to figure that out before I can build the calc.

20. Mar 27, 2009

### Ike47

Again, thank you. Perhaps I should be able to see this by myself, but could you clarify how to calculate t(earth) from these formulas, both for the acc/deceleration stages and for the coasting stage? t in your formulas is t(vehicle), or so I understand, right? During coasting, is t(earth) the simple d/v calculation, or does that number have to be adjusted by the inverse of the time dilation factor?