Math of relativistic space travel

[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 ). 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.

 

[DaveC426913]

Hm. I keep needing this too.

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

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

 

[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.

 

[DaveC426913]

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.

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.)

 

[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...

 

[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...

 

[DaveC426913]

...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!

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.

 

[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.

 

[DaveC426913]

At 1g, you could make it to the Andromeda galaxy in 30 years!

http://musr.physics.ubc.ca/~jess/p200/str/str13.html

 

[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.

To address one other question,

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.

 

[DaveC426913]

I'm not sure that 'an asteroid's worth' of fuel would help that much, since the increased mass may be self-defeating.

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...

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,

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.

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.

The amount of fuel does not play into 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.)

 

[Ike47]

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...

I think I follow this, but if the needed fuel for a constantly accelerating and decelerating trip is impractical (a small asteroid's worth), then wouldn't a trip with a long middle 'coasting' component be more feasible? However, I have no idea how to work out what the 'most feasible' amount of fuel and percentage of coasting time would be.


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.

I guess I didn't make myself clear. In my example, I'm not trying to get the vehicle back to earth, just get it to its destination and be able to stop there (where it can presumably either colonize, or gather more fuel to go somewhere else). So I think we are in agreement on this point. I don't follow your comment about the 10T payload however. Are you saying that the payload is too small to achieve high Earth orbit (if so, I don't understand), or that the 10:1 fuel:payload ratio is insufficient?



The amount of fuel does not play into 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.)

I think the amount of fuel plays into the feasibility of constructing and sending the vehicle. Perhaps the best first step would be to consider what we could do today, if cost were not a constraint. Let us say we build a vehicle in modules, send them into high Earth orbit separately, put the vehicle together there, and send the fuel to it in small increments. If we assume an arbitrary payload mass, say 10T or 100T, and if we assume a maximum feasible fuel:payload ratio of, e.g., 100:1, given a currently constructable controlled fission engine, what would be the quickest flight plan to get to a destination of 4 or 10 or however many ly? A constant acceleration and deceleration (presumably much less than 1g), or a high but relatively brief acceleration and deceleration, with a long coasting period in between? Does this method of attacking the question make sense?

Hope this is readable. :)

 

[Janus]

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.

To address one other question,

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.

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
V_e 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

 

[DaveC426913]

I don't follow your comment about the 10T payload however. Are you saying that the payload is too small to achieve high Earth orbit (if so, I don't understand), or that the 10:1 fuel:payload ratio is insufficient?

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.

 

[Ike47]

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
V_e 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

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)?

 

[DaveC426913]

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

 

[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.

 

[Janus]

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)?

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.

 

[DaveC426913]

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.

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.

 

[Ike47]

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.

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?

 

[DaveC426913]

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

You see why I want to build a calculator...

 

[DaveC426913]

t is the time
d in the distance traveled
v is the velocity attained
a is the acceleration
c is the speed of light
V_e 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

Does the exhaust velocity limit the upper velocity? Why can the rocket not accelerate all the way to the halfway point?

 

[DaveC426913]

I found a calculator.

http://www.cthreepo.com/cp_html/math1.htm
Go down to 'Long Relativistic Journeys'

 

[Ike47]

I found a calculator.

http://www.cthreepo.com/cp_html/math1.htm
Go down to 'Long Relativistic Journeys'

Lol, I mentioned that in my first post. As I noted there, it doesn't give the formulas, nor does it include the option of any coasting.

 

[DaveC426913]

Lol, I mentioned that in my first post. As I noted there, it doesn't give the formulas, nor does it include the option of any coasting.

Oops.

Well, the formulae are exposed in the source code, and the coasting time shouldn't be too hard to work in.

But seriously, do you need the formulae if you have a calculator?

 

[Janus]

Does the exhaust velocity limit the upper velocity? Why can the rocket not accelerate all the way to the halfway point?

The exhaust velocity doesn't but the mass ratio does. Since I was assuming a starting mass ratio of 110, you can only accelerate to the point where you just have enough fuel left to decelerate upon arrival. You could adjust the acceleration so that you reached the mid way point at just that moment, but the trip would take a lot longer.

 

[Janus]

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?

To get Earth time for the acceleration phases you use

T_{earth} = \frac{c}{a} \sinh \left ( \frac {a t_{ship}}{c} \right )


During coasting you use the d/v formula for Earth time, and apply the time dilation formula to this to get the ship time.

 

[Ike47]

To get Earth time for the acceleration phases you use

T_{earth} = \frac{c}{a} \sinh \left ( \frac {a t_{ship}}{c} \right )


During coasting you use the d/v formula for Earth time, and apply the time dilation formula to this to get the ship time.

Thank you yet again. I'm going to be rather busy this weekend, but after that I'll try a calculation including coasting, using arbitrary numbers, and see if I can get the desired values from the formulas. Once I post them, I'd much appreciate it if you'd look them over to see if they seem correct.

 

[Ike47]

Thank you again, Janus. Obviously I was a bit optimistic when I thought I'd be ready to work on this by last week. But I have finally made some progress, coming up with 'statistics' for my vehicle (mass, mass ratio, total energy available from fuel, etc.), and as I try to apply these parameters to the formulas you gave, I come up with two questions:

1. If I know the total energy output of my vehicle (total mass or weight of my fuel, times the energy output (via fusion in this case) per mass or weight unit), how do I calculate exhaust velocity? (Also, if this requires a value for the nozzle, as I know specific impulse does, what is a reasonable value for that?)

2. (I'm not quite sure I'm asking the question properly in this instance, but I think it's related to the question by DaveC about the limit to the upper velocity.) Since I want to include a period of 'coasting', since I (presumably) don't have enough fuel to accelerate and decelerate during the whole trip, and since I need to know the speed attained when I start to coast, how do I determine at what point the vehicle has to turn off its engine (and later restart it) to achieve maximum effectiveness?

If the maximum effectiveness occurred by accelerating and decelerating for equal distances, the answer would be easy, but as I have read elsewhere, and as you answered to DaveC, that is not the case because of the mass ratio. Specifically, you have to use more fuel accelerating because you have a higher mass to accelerate, while during deceleration, your initial mass is much lower than the initial mass of accelerating, and likewise for your final mass at the end of accelerating and decelerating.

So, if I understand all this correctly, how do I determine time and distance for the accelerating part of the trip and for the decelerating part of the trip, so I can also calculate the time and distance for the coasting part of the trip?

3. (Ok, I just remembered a third question, just a clarification really.) Am I correct in assuming that your formulas are based on starting the vehicle trip at (essentially) 0 gravity? In other words, it is assumed that the vehicle does not have to escape Earth's gravity well?

Thank you again for your help and patience.

 

[JesseM]

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.

There is another option you haven't considered: instead of a ship bringing all the needed fuel with it (adding greatly to its initial mass), the ship might be pushed along to its destination by something like a high-powered laser located in our solar system (with the ship having a large solar sail to collect the photons even as they spread out since no laser can be perfectly collimated) or by pellets shot out at high speed by something like a large mass driver in the solar system (the advantage of pellets is that they could have some small supply of fuel which would allow them to make small course corrections to keep the 'beam' of pellets collimated). Both of these are discussed in http://www3.interscience.wiley.com/journal/118692597/abstract by Paul Gilster, a short paper on the feasibility of different methods of interstellar travel--I can send you a copy if you're interested, just send me a PM with your email.

 

[Ike47]

There is another option you haven't considered: instead of a ship bringing all the needed fuel with it (adding greatly to its initial mass), the ship might be pushed along to its destination by something like a high-powered laser located in our solar system (with the ship having a large solar sail to collect the photons even as they spread out since no laser can be perfectly collimated) or by pellets shot out at high speed by something like a large mass driver in the solar system (the advantage of pellets is that they could have some small supply of fuel which would allow them to make small course corrections to keep the 'beam' of pellets collimated). Both of these are discussed in http://www3.interscience.wiley.com/journal/118692597/abstract by Paul Gilster, a short paper on the feasibility of different methods of interstellar travel--I can send you a copy if you're interested, just send me a PM with your email.

Thanks for the reference. While it sounds like an article I'd really like to read, I've looked at the link and it seems there is a fee to see it, or presumably to download it. I don't believe in trying to circumvent copyright laws (e.g., taping something on TV is fine, but downloading the item from a pirate internet site is not, for me), so unless this can be done 'legally' I will pass on PM'ing you my email address. Thank you for the offer though!

 

[JesseM]

Thanks for the reference. While it sounds like an article I'd really like to read, I've looked at the link and it seems there is a fee to see it, or presumably to download it. I don't believe in trying to circumvent copyright laws (e.g., taping something on TV is fine, but downloading the item from a pirate internet site is not, for me), so unless this can be done 'legally' I will pass on PM'ing you my email address. Thank you for the offer though!

In most cases with scholarly papers, the authors are happy to email copies to people for free even if the journals that published them have a standard fee for downloading content; the author of this paper is Paul Gilster, who writes the blog Centauri Dreams, so you could also try dropping him a line at his email here and ask if he'd be willing to send you a copy.

 

[Ike47]

Thank you again! I've tried to find out the price of purchasing an online copy, and whether I can print it for my personal use, from the link location, Wiley... no luck so far, but I've submitted queries. Once I get answers, I'll be happy to purchase the 8 page article if the price is nominal; otherwise (or if they won't respond), then I'll try contacting the author. I really appreciate your assistance on this.

 

[qraal]

Hi All

Ike47, have you decided what the cruising speed is? There are pretty simple relativistic equations for the total trip time from a planet reference frame for a boost-coast-boost flight-path for constant acceleration flight. I worked them out to analyse a couple of stories some years ago, then discovered the cruise speed was too low for it to make much difference.

There are also equations for constant thrust trajectories, which are quite complex to make relativistic, especially if you assume several stages with different thrusts. For example, the boost-phase of the "Daedalus" starprobe involves two stages and constant thrust, but since the final cruising speed is just 0.122 c the relativistic result isn't different by more than a couple of %.

At extremely high speeds close to light there's interstellar drag that must be factored in, but we're talking about flying at time dilation factors of thousands or more. That would require technology beyond anything foreseeable.

So what's your cruising speed or range of speeds? Power source? Acceleration range?

 

[DaveC426913]

At extremely high speeds close to light there's interstellar drag that must be factored in

Can you elaborate on this "interstellar drag"?

 

[Ike47]

Hi, Graal. I'm still waiting and hoping to get answers from Janus about calculating exhaust velocity and about how to determine when coasting should start in an accelerating-coasting-decelerating scenario.

I want to be able to vary the inputs to see the effect on flight time (both ship referenced and Earth referenced), but the initial set of parameters I came up with, assuming (100% efficient) fusion of hydrogen, and using the statistics of the Saturn V rocket as a starting point were:

M_T = 2.3 x 10⁶kg

MR = 4.6

Total energy: 1.134 x 10²¹ J

(This last based on 2.6 x 10⁴ m³ of liquid hydrogen, with a mass of 1.8 x 10⁶ kg, and an energy output of 6.3 X 10¹⁴ J kg^-1.)

If I could convert that energy to an exhaust velocity, I could solve the equations Janus suggested. Then I'd just have to be able to determine when the ship starts coasting, since its coasting speed will depend on the ideal time to shut down the engines.

As for interstellar drag, DaveC, I think he's referring to the particles in 'empty' interstellar space, which I believe are thought to be in the vicinity of one atom per cubic meter. While that amount of contact with a ship is negligible at ordinary speeds, if the ship is at relativistic speed, then the mass of the those particles becomes effectively great enough to slow down the ship by drag... if they don't destroy it instead. :)

 

[qraal]

Can you elaborate on this "interstellar drag"?

Same as regular drag, but it depends on how the starship handles it. If using a magnetic field to deflect ions, then that will create drag over the whole effective area of the field's influence (which I'm not sure exactly how to calculate, though I have a rough idea.)

The other question is whether any kind of aerodynamic reduction of the drag is feasible when handling a very high flux of interstellar ions, atoms and dust. Otherwise the incident flux on the frontal cross-section of a vehicle will be very high.

To illustrate, the Interstellar Medium (ISM) is typically quoted at being about 3 Jupiter masses per cubic light-year, and is composed largely of individual hydrogen atoms. It works out at, on average, about 1 million protons per cubic metre (it's lower than average near the Sun's part of the Galaxy) or ρ = 1.67x10^-21 kg/mγ. Crashing through the protons and absorbing their energy directly means the frontal incident flux is ~1/2ρv³ per square metre. That's a good estimate at low speeds < 0.5 c. Once you're traveling at relativistic speeds the γ factor appears in the equations and it gets a bit more elaborate mathematically. First the kinetic energy of the incident protons becomes m_pc²(γ-1), and the volume traveled through by each square metre of frontal area per second is (vγ) m³ (in the ship's reference frame.) Thus the incident energy is now E = ρc²(γ-1)(vγ) J/s which means it gets really, really hot in a hurry.

 

[qraal]

Hi Ike47

It's Qraal, which sounds a bit like my real name.

I worked out the equations for a boost-coast-brake flightplan, both classical and relativistic. Just let me know if you want them still.

As for your exhaust velocity computations it's usually best to use 'c' units and work out the energy in terms of mass ratios from before and after the reaction. For example, straight hydrogen fusion like the Sun does, has a net energy production of 0.007 relative to the mass of the original hydrogens. That means the resulting helium's energy-to-mass ratio is 0.007/(1-0.007) = 0.00705 and the resulting gamma-factor is that plus 1 i.e. 1.00705. Invert it, square it and subtract it from 1, then take the square root to get the velocity of the exhaust in units of 'c' (lightspeed.)

i.e. v = (1-γ^-2)^0.5

which in this case gives v = 0.118 c.

But - a big 'but' - that's an ideal case using a non-ideal reaction. Proton-fusion is slow and hard to start. A proton at 15.7 million K and 215 billion bars of pressure (i.e. inside the middle of the Sun) has a 50% probability of reacting with another proton in 9 billion years. Slow. Not good for rockets. Proton-proton reactions don't speed up much until about 10 billion degrees or more, a temperature we just can't contain at sufficient density.

A good place to find fusion reaction details is here...

http://www.ibiblio.org/lunar/school/InterStellar/Explorer_Class/Bussard_Fusion_systems.HTML" (forgive the bad spelling)

If you want to be especially precise you can look up all the reactions in detail around the web and use precise isotope masses to work things out. However in the real world fusion engines will get less than ideal performance.

Hi, Graal. I'm still waiting and hoping to get answers from Janus about calculating exhaust velocity and about how to determine when coasting should start in an accelerating-coasting-decelerating scenario. [snipped]

If I could convert that energy to an exhaust velocity, I could solve the equations Janus suggested. Then I'd just have to be able to determine when the ship starts coasting, since its coasting speed will depend on the ideal time to shut down the engines.

 

[Ike47]

Thanks, Qraal, and sorry for misreading your name.

First, just to respond about interstellar drag, your numbers are right, but probably overpessimistic. (My number, 1 atom m^-3, applies to intergalactic space, e.g., Smoot and Davidson, Wrinkles in Time, p. 158.) I say overpessimistic, in that the 1 million protons m^-3 is an average, and by avoiding dust clouds, nebulae, etc., one should manage with about a few thousand atoms or protons m^-3, at least according to the only partially documented Wikipedia page on interstellar medium.

Mostly I wanted to respond to your comments and offer about space travel formulae, however. Yes, I'd very much like to see the formulae you've calculated for acc-coast-dec space flight. Especially because I'd like to be able to vary the input data and see the effect on the time various destinations would require.

Thanks as well for the exhaust velocity formula. I quite understand that 100% energy conversion is impossible, but it provides a rough estimation. Plus, assuming unknown scientific and engineering advances in, say, the next 500-1000 years, perhaps 90-95% efficiency might be possible. Or not. :)

I didn't try to maximize all my data idealistically, btw. For example, I just used H to He, rather than the slightly more powerful H to Fe (which, from your formula, I calculate would create a v of 0.13 c). Nor did I assume a compressed gaseous state for the hydrogen, as in the Sun, but only liquid hydrogen. So I figured assuming 100% energy conversion efficiency wouldn't be too far off, hypothetically. Of course, the controlled fusion of liquid hydrogen within the confines of a rocket engine is fantasy at this point anyway, but that's why I'm talking about using this for science fiction, not serious speculation on the possibilities of interstellar flight.

Another complication I thought of is that efficient fusion of H might require a catalyst, as in the CN cycle of H to He. That would further worsen the vehicle's mass ratio. (This, and most of my information on fusion, comes from a book I've found very useful, Tayler's The Stars: Their Structure and Evolution, 1994 edition.)

In any case, with the formula you provided for exhaust velocity, I think I have everything I need except the formulas for acc-coast-dec (or boost-coast-brake). I'm guessing that these formulas would be similar to the ones Janus provided earlier in this thread, except that they would allow the calculation of the coasting beginning and ending points (time, distance), so the coasting velocity could be known. If you could provide those formulae (it sounds like you've already calculated them), I would really appreciate it!

 

[DaveC426913]

... by avoiding dust clouds, nebulae, etc.,

Have you considered the implications of trying to dodge obstacles with your ship in terms of trip-time - but more significantly - fuel?

It could easily double your fuel requirements. Say there were a mere two dust clouds blocking your path to your destination and the shortest route was between them. You'd have to accelerate your ship to relativistic velocities perpendicular to your path to do so. In other words, you've just added a whole trip within a trip fuel-wise.

It's entirely possible it would be more efficient to add a shield simply plow through the clouds.

 

[Ike47]

True, DaveC. But if your object was to explore the galaxy, or at least the relatively nearby stars in it (for example, all the stars and possible associated planets within, say, 50 parsecs of the Sun), I'd think you could find target stars from Earth with much less intervening matter than other target stars. You could try going to those first, and find paths that had low interstellar density from those stars to the stars that weren't good destinations from Earth directly.

In this manner, I would think you could get to most of the stars within any given region of the galaxy more efficiently than just 'plowing through' whatever density you might come upon. Of course, all this assumes you could measure the interstellar density at any given location reasonably accurately from Earth. To what degree that is, or may become true, I don't know.

 

[DaveC426913]

True, DaveC. But if your object was to explore the galaxy, or at least the relatively nearby stars in it (for example, all the stars and possible associated planets within, say, 50 parsecs of the Sun), I'd think you could find target stars from Earth with much less intervening matter than other target stars.

I would hope that the destination would be deciding factor, not the ease of the journey.

"Hey, this star has an Earth-like planet with an oxygen atmo!"
"Yeah, but look all thst dust in the way... let's go for that dead iceball instead."
"Uh, this is my entire life's work hangin' in the balance..."

 

[qraal]

Hi All

These equations are for boost-cruise-brake trajectories with constant acceleration, a; displacement, S; travel-time (planet reference frame) t, and speed, v, with γ = (1-(v/c)²)^-1.

Classical: a = c²/(vt - S)*((γ² - 1)/γ²)

Relativistic: a = c²/(vt - S)*((γ - 1)/γ)

...which is how I used the equations originally. I put the classical version in terms of γ (and v) to compare the two equations more readily - naturally it has no physical meaning in classical dynamics. A simpler version for the classical situation is:

t = v/a + S/v

which becomes...

a = v²/(vt - S)

...the two are related by the factor a_r/a_c = 2γ/(γ + 1), where a_r is the relativistic acceleration needed for the same v in time t, and a_c is the classical acceleration.

From the ship-board point of view the travel time, τ, is a bit more complicated by a cosh(γ) factor. Here's how it goes...

τ_a = 2c/a*arcosh(γ)

τ_c = (S - S_a)/vγ

thus τ = τ_a + τ_c

where τ_a is the total boost/brake time, and τ_c is the coast time. S_a is the total displacement while boosting/braking. In terms of S_a, a & c the γ factor is 1+aS_a/2c². From there you can infer the rest.

As for constant thrust dynamics, the relativistic equations are quite elaborate, but the difference between them and the classical result is minor. One problem with big mass-ratios and a constant thrust is that the average velocity ends up much less than the exhaust velocity over the acceleration track. That can be true for constant acceleration too, but it's worse for constant thrust. Better performance comes from constant acceleration and dropping off spent fuel-tanks and engines as you go, or using them for propellant (which you can do with lithium for example.)